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CHEMICAL 
CALCULATION   TABLES 


FOR  LABORATORY  USE 


BY 

HORACE   L.    WELLS 

Professor  of  Analytical  Chemistry  and  Metallurgy  in  the 
Sheffield  Scientific  School  of  Yale  University 


SECOND    EDITION,    REVISED 

WITH 

A  DOUBLE  THUMB-INDEXED 
LOGARITHM  TABLE 


NEW  YORK 

JOHN    WILEY    &   SONS,    Inc. 

London:    CHAPMAN    &    HALL,    Limited 

1919 


Copyright,  1903, 

BY 

HENRY  HOLT  &  CO. 


Copyright,  1919, 


HORACE  L.  WELI^ 


Stanbopc  pctss 

F.    H.  GILSON  COMPANY 
BOSTON,  U.S.A. 


PREFACE 

This  book  of  tables  for  every  day  use  in  the  chemical  laboratory  is  a  revision 
and  modification  of  my  ''Tables  for  Chemical  Calculations"  published  about  16 
years  ago,  and  now  out  of  print.  The  changes  in  our  official  atomic  weights 
have  made  necessary  a  complete  revision  of  most  of  the  tables.  A  new  table  for 
facilitating  the  calculation  of  percentage  compositions  of  organic  compounds  has 
been  added,  while,  in  order  to  make  the  book  more  compact  and  thus  more  con- 
venient for  its  chief  uses,  several  of  the  smaller,  less  important  tables  have  been 
omitted,  while  the  matter  relating  to  the  explanation  of  the  tables  and  the  illus- 
tration of  their  uses  has  been  either  left  out  or  much  changed  and  abbreviated. 

The  table  of  logarithms  of  numbers  has  been  provided  with  a  double  thumb- 
index  which  is  believed  to  be  a  novel  and  useful  device.  With  this  improvement 
one  can  turn  from  any  place  in  the  table  to  any  other  reference,  backward  as 
well  as  forward,  by  practically  a  single  motion,  since  all  indices  are  visible  from 
every  part  of  the  table. 

The  greatest  care  has  been  used  to  secure  accuracy  in  the  data  by  the  duplica- 
tion, at  least,  of  their  derivation,  with  a  complete  verification  after  the  plates 
were  cast. 

There  is  no  doubt  that  accurate  calculation  is  exceedingly  important  in  con- 
nection with  quantitative  analysis  and  other  kinds  of  chemical  work,  and  it  is 
believed  that  these  tables  will  be  of  assistance  in  avoiding  errors  of  calculation, 
as  well  as  in  making  the  work  more  rapid  and  less  laborious  than  is  often  the  case. 

H.  L.  W. 


CONTENTS 

PAOB 

Introduction 1 

TABLE 

I.  Atomic  Weights 3 

II.   Gravimetric  Factors 4 

III.  Formula- Weights 11 

IV.  Indirect  Analyses 17 

V.  Reduction  of  Gas- Volumes  to  O''  and  760  mm 18 

VI.   Calculation  of  Percentage  of  Nitrogen  from  the  Gas- Volume  19 

VII.  Barometer  Corrections  for  Temperature 20 

VIII.   Multiples  for  Organic  Compounds 21 

IX.   Constants  for  Molecular- Weight  Determinations 22 

X.  Weights  and  Measures 23 

Notes  on  the  Use  of  Logarithms 24 

XI.  Logarithms  of  Numbers 25 


CHEMICAL    CALCULATION    TABLES 


INTRODUCTION 

Five-place  logarithms  have  been  adopted  in  this  book  because  they  are  satis- 
factory in  usually  introducing  no  inherent  errors  of  calculation  into  the  results 
obtained  from  the  four-figure  and  five-figure  numbers  frequently  used  as  data  in 
chemical  computations. 

It  is  true  that  four-place  logarithms  will  give  amply  accurate  results  for  a 
great  deal  of  analytical  work,  but  as  these  are  liable  to  give  incorrect  fourth  figures 
in  some  cases  the  longer  logarithms  are  to  be  preferred  for  general  use.  It  is 
easy  to  use  these  with  the  conveniently  indexed  table  here  supplied,  and  they  have 
the  additional  advantage  that,  since  usually  only  four  figures  at  the  most  are 
significant  in  the  results  of  analytical  work,  these  can  be  taken  from  the  nearest 
logarithm  without  the  trouble  of  interpolating  for  a  fifth  figure. 

Computers  should  appreciate,  and  apply  in  carrying  out  their  figures,  the 
effects  upon  their  results  of  the  apparent  errors  of  the  approximate  numbers 
employed.  For  general  guidance  in  this  matter  the  following  rule  may  be  given: 
Take  as  a  basis  the  smallest  number  of  apparently  reliable  figures  in  any  of  the 
approximate  numbers  used  by  multiplication  and  division  (not  including  addi- 
tion and  subtraction)  in  producing  the  result.  Then,  if  the  number  used  for  the 
basis  begins  with  a  smaller  figure  than  does  the  result,  the  number  of  figures  to  be 
preserved  in  the  result  will  usually  be  equal  to  the  basis,  while  if  the  reverse  is  true 
in  regard  to  the  first  figures  the  number  of  figures  to  be  considered  significant  in 
the  result  will  usually  be  one  more  than  the  basis.  It  should  be  borne  in  mind 
that  zeros  at  the  end  of  decimals  may  be  apparently  reliable.     For  example: 

a.  23.00--       X  2.5417 --       =  58.46--        (not  58.5  nor  58.459) 

b.  0.0089—     X  0.16598 —     =  0.00148--     (not  0.0015  nor  0.001475) 

c.  0.2301  -  -     -^  2.5600  -  -       =  0.08988  -  -     (not  0.0899  nor  0.89883) 

This  rule  is  by  no  means  an  absolute  one,  and  it  should  be  modified  according 
to  judgment.  For  instance,  in  the  example  d  given  above,  since  88  is  so  much 
larger  than  the  18  from  which  it  is  derived,  it  might  well  be  preferred  to  call  the 
result  0.09%.  In  some  cases  it  is  desirable  to  preserve  a  very  uncertain  last  figure, 
while  in  other  instances  it  is  best  to  omit  it,  but  it  is  very  bad  practice  to  report 
more  than  one  evidently  uncertain  figure,  or  to  leave  off  or  fail  to  calculate  one 
or  more  apparently  accurate  ones.  Attention  may  be  called  here  to  the  im- 
portance of  adding  a  unit  to  the  preceding  figure  when  a  figure  greater  than  5  is 
left  off. 

1 


2  CHEMICAL  CALCULATIONS 

For  a  more  extensive  discussion  of  the  use  of  approximate  numbers  in  calcu- 
lations, the  author's  ''Text  Book  of  Chemical  Arithmetic"  may  be  referred  to. 

In  order  that  accidental  errors  of  calculation  may  be  avoided  it  is  recommended 
that  entirely  independent  duphcations  of  all  calculations  should  be  carried  out, 
for  mistakes  are  not  always  discovered  by  simply  looking  over  the  work. 

In  Tables  II  and  III  the  logarithms  are  generally  given,  for  the  sake  of  uni- 
formity, as  though  the  atomic  weights  were  exact,  instead  of  approximate,  num- 
bers. For  instance,  a  five-place  logarithm  is  given  for  the  atomic  weight  of  boron, 
ILO — ,  just  as  though  it  were  known  to  be  11.000.  This  feature  need  cause 
no  trouble  as  far  as  the  gravimetric  factors  are  concerned,  since  the  latter  are 
practically  always  more  accurate  than  the  data  to  which  they  are  appUed,  and 
hence  they  need  not  be  considered  in  deciding  upon  the  significant  figures  of  the 
results  of  ordinary  calculations. 

The  numerical  formula  weights  in  Table  III  in  some  cases  have  an  uncertain 
last  figure. 

Hg     200.6 
For  example,  in  the  addition        S        32.06 

232.66 

the  last  figure  in  the  sum  is  uncertain  because  it  has  been  obtained  by  addition  to 
an  unknown  (plus  or  minus)  figure.  It  has  been  preferred  in  most  cases  to  retain 
such  uncertain  figures,  since  leaving  them  off  and  giving,  for  instance,  HgS  as  232.7 
would  be  Uable  to  decrease  the  accuracy  of  such  formula- weights.  In  using  these 
numbers  for  calculations  it  will  usually  make  no  difference  whether  or  not  the  final 
figures  are  apparently  accurate,  but  a  glance  at  the  atomic  weights  will  show 
which  is  the  case. 

The  student  should  observe  that  many  of  the  numerical  factors  in  Table  II, 
when  multiplied  by  100,  give  percentages  of  constituents,  and  that  they  may  often 
be  conveniently  used  in  this  way.  For  example,  NaCl  — >  Na  =  0.39444  shows 
that  NaCl  contains  39.44%  of  Na  and  60.56%  of  CI.  Since  the  factor  0.39444  is 
derived  from  23.00--  ,  its  last  figure  is  uncertain  and  has  not  been  used  in 
giving  the  percentages  (compare  the  rule  previously  given) . 

It  should  be  stated  that  the  value  0.0012510  g.  was  adopted  in  Table  VI  as  the 
weight  of  1  cc.  of  nitrogen  gas  at  41°  latitude,  instead  of  the  value  0.0012514  g. 
at  45°  because  the  former  latitude  is  more  suitable  for  general  reference  in  the 
United  States,  but  that  the  change  makes  a  difference  of  only  about  one  part  in 
3000  and  is  entirely  inappreciable  in  nitrogen  determinations  by  the  Dumas 
method. 


ATOMIC  WEIGHTS 


Table  I. 

ATOMIC  WEIGHTS/ 

O  =  16. 


Aluminium 

Antimony 

Argon 

Arsenic 

Barium 

Beryllium  (Glucinum) . . . 

Bismuth 

Boron 

Bromine 

Cadmium 

Caesium 

Calcium 

Carbon . . .': . . . .,,. ... . .  ^^*-, 

Cerium .^' 

Chlorine 

Chromium 

Cobalt 

Copper 

Dysprosium 

Erbium 

Europiimi 

Fluorine 

GadoUnium 

GalUum 

Germanium 

Gold 

Helium 

Holmium 

Hydrogen 

Indium 

Iodine 

Iridium 

Iron 

Krypton 

Lanthanum 

Lead 

Lithium 

Lutecimn 

Magnesiima 

Manganese 

Mercury 

Molybdenum 


AI 

27.1 

Sb 

120.2 

A 

39.88 

As 

74.96 

Ba 

137.37 

Be 

9.1 

Bi 

208.0 

B 

11.0 

Br 

79.92 

Cd 

112.40 

Cs 

132.81 

Ca 

40.07 

,Q 

12.005 

Ce 

140.25 

CI 

35.46 

Cr 

52.0 

Co 

58.97 

Cu 

63.57 

Dy 

162.5 

Er 

167.7 

Eu 

152.0 

F 

19.0 

Gd 

157.3 

Ga 

69.9 

Go 

72.5 

Au 

197.2 

He 

4.00 

Ho 

163.5 

H 

1.008 

In 

114.8 

I 

126.92 

Ir 

193.1 

Fe 

55.84 

Kr 

82.92 

La 

139.0 

Pb 

207.20 

Li 

6.94 

Lu 

175.0 

Mg 

24.32 

Mn 

54.93 

Hg 

200.6 

Mo 

96.0 

Neodymiimi Nd  144.3 

Neon Ne       20.2 

Nickel Ni        58.68 

Niobium  (Columbium) . .  Nb       93 . 1 

Niton Nt  222.4 

Nitrogen N         14.01 

Osmium Os  190.9 

Oxygen O         16.00 

Palladium Pd  106.7 

Phosphorus P         31.04 

Platinum Pt  195.2 

Potassium K         39.10 

Praseodymium . . . . . Pr  140 . 9 

Radium Ra  226.0 

Rhodium Rh  102.9 

Rubidium Rb       85.45 

Ruthenium .  '.^.C . ^ t*?f  .r^Rii  101 . 7 

Samarium  .^ffi-  .*■  .^.^ot .-?  yA/^m  150 . 4 
Scandium . . H^** ."^ . y t"  ypyT  Sc  44 . 1 
Selenium  ...i^l^  rfK><K .  Se        79 . 2 

Sihcon Si         28.3 

Silver Ag  107.88 

Sodium Na       23.00 

Strontium Sr        87.63 

Sulphur S  32.06 

Tantalum Ta  181.5 

Tellurium Te  127.5 

Terbium Tb  159.2 

ThalHum Tl  204.0 

Thorium Th  232.4 

Thulium Tm  168.5 

Tin Sn  118.7 

Titanium Ti        48.1 

Tungsten W  184.0 

Uranium U  238.2 

Vanadium V         51.0 

Xenon X  130.2 

Ytterbium Yb  173.5 

Yttrium Yt       88.7 

Zinc Zn        65.37 

Zirconium Zr        90 . 6 


This  is  the  latest  International  Table. 


GRAVIMETRIC  FACTORS 


Table  II. 
GRAVIMETRIC  FACTORS. 


Given. 

Sought. 

Multiply  by  Factor. 
N.                      Log. 

Aluminium,  27.1 

AI2O3 
AIPO4 

AI2 
AI2O3 

(i)Al203 

.5303 
1.886 
.4184 

.72  455 
.27  545 
.62  156 

Antimony,  120.2 

Sb204 
Sb204 
Sb204 

Sb2 

Sb203 

Sb205 

.7898 

.9475 

1.0526   ' 

.89  749 
.97  656 
.02  225 

Sb2S3 
Sb2S3 
Sb2S3 

Sb2 

Sb203 

Sb205 

.7142 
.8569 
.9519 

.85  384 
.93  291 
.97  860 

Arsenic,  74.96 

Sb2 

Sb2 

AS2S3 

Sb203 

SbzOs 

As2 

1 . 1997 

1.3328 

.6092 

.07  907 
.12476 

.78475 

AS2S3 

AS2S3 
AS2S3 

AS203 
AS205 

(2)As04 

.8042 

.9343 

1 . 1293 

.90  538 
.97  047 
.05  281 

AS2S5 

Mg2AS207 
Mg2AS207 

AS2 

AS2 

AS2O3 

.4833 
.4827 
.6373 

.68  419 
.68  372 
.80  435 

Mg2AS207 
Mg2AS207 

Ag3As04 

AS2O5 

(2)As04 
As 

.7403 
.8949 
.1620 

.86  944 
.95  178 
.20  962 

Barium,  137-37 

AS2 

As2 
BaS04 

AS2O3 
AS2O5 
Ba 

1.3202 
1.5336 

.5885 

.12  063 

.18572 
.76  973 

BaS04 

BaCr04 

BaCr04 

BaO 

Ba 

BaO 

.6570 
.5422 
.6053 

.81  758 

•73  413 
.78  198 

BaC03 
BaCOa 
Ba 

Ba 

BaO 

BaO 

.6960 

.7770 

1.1165 

.84  260 
.89  045 
.04  785 

Bismuth,  208.0 

BiaOa 

BiOCl 

BiOCl 

Bi2 

Bi 

(i)Bi203 

.8965 
.8017 

.95  257 
.90  399 

.8942 

.95  142 

GRAVIMETRIC  FACTORS 


Given. 

Sought. 

Multiply  by  Factor. 
N.                    Log. 

BisSs 
BigSs 

Bi2 

Biz 

BizOa 

BizOs 

.8122 

.9059 

1.1154 

.90  967 

.95710 
.04  743 

Boron,  ii.o 
Bromine,  79.92 

B203 
B2 

AgBr 

B2 

B2O3 
Br 

.3143 
3.182 
.42556 

.49  732 
.50  268 
.62  896 

AgBr 
Br— CI 
Br— CI 

HBr 

Br 

AgBr 

.43092 
1.7976 
4.2241 

.63  440 
.25  469 
.62  573 

Cadmium,  112.40 

Br 

CdO 
CdS 

(i)0 

Cd 

Cd 

.1001 
.8754 

.7781 

.00  043 
.94  220 
.89  102 

Caesium,  132.81 

CdS 

Cd 

CS2SO4 

CdO 
CdO 

CS2 

.8888 

1 . 1424 

.7344 

.94  882 
.05  780 
.86  593 

Calcium,  40.07 

Cs2PtCl6 
CS2 

CaO 

CS2 

CS2O 
Ca 

.3943 
1.0602 
.71465 

.59  587 
.02  540 

.85  409 

CaO 

CaS04 

CaS04 

CaCOa 

Ca 

CaO 

1.7848 
.2944 
.4119 

.25  159 
.46  886 

.61  477 

CaCOs 
CaCOa 
Ca 

Ca 

CaO 

CaO 

.4004 

.5603 

1.3993 

.60  250 
.74  840 
.14591 

Carbon,  12.005 

Ca 

CaO 

BaCOs 

CaCOs 
CaS04 
CO2 

2.4975 

2.4279 

.2229 

.39  750 
.38  523 
.34  820 

CaCOs 

CO2 

C 

CO2 

C 

CO2 

.4397 

.2728 

3.6656 

.64318 
.43  586 
.56414 

Chlorine,  35.46 

CO2 

AgCl 

AgCl 

CO3 

CI 

HCl 

1.3636 

.24738 
.2544 

.13  468 
.39  337 
.40  555 

Ag 
CI 
AgCl 

CI 

(4)0 

(1)0 

.3287 
.2256 
.05581 

.51  680 
.35335 
.74  672 

GRAVIMETRIC  FACTORS 


Given. 

Sought. 

Multiply  by  Factor. 
N.                    Log. 

Chromium,  52.0 

Cr203 
Cr203 
PbCr04 

Cr2 

(2)Cr03 

Cr 

.6842 

1.3158 

.1609 

.83  519 
.11919 
.20  653 

PbCr04 
PbCr04 
Cr2 

(i)Cr203 

Cr03 

Cr203 

.2351 
.3094 
1.462 

.37  134 
.49  053 
.16481 

Cobalt,  58.97 

Cr 

C0SO4 

C03O4 

Cr03 
Co 

C03 

1.923 
.3804 
.7343 

.28  400 
.58  022 
.86  589 

Copper,  63.57 

Co 

CuO 

Cu 

CoO 

Cu 

CuO 

1.2713 

.7989 

1.2517 

.10  426 
.90  250 
.09  750 

CU2S 
CU2S 
CuSCN 

CU2 

(2)CuO 
Cu 

.7986 
.9996 
.5226 

.90  234 
.99  984 
.71815 

Cyanogen,  26.015 

CuSCN 

AgCN 

Ag 

CuO 

CN 
CN 

.6541 
.1943 
.2411 

.81  565 
.28  846 
.38228 

Fluorine,  19.0 
Hydrogen,  1.008 

CaF2 

SiF4 

H2O 

F2 
F4 
H2 

.4867 
.7287 
.1119 

.68  730 
.86  253 
.04  884 

Iodine,  126.92 

Agl 

Pdl2 

I— CI 

I 

I2 

I 

.5405 

.7041 

1.3877 

.73  283 
.84  760 
.14230 

Iron,  55.84 

I— CI 
Fe203 
FeaOa 

Agl 
Fe2 

(2)FeO 

2.5672 
.6994 
.8998 

.40  947 
.84  473 
.95415 

FezOa 

FeS 
Fe 

(2)FeS2 

Fe 

FeO 

1.5025 

.6353 

1.2865 

.17  682 
.80  296 
.10  942 

Lead,  207.20 

Fe2 

PbS04 

PbS04 

Fe203 

Pb 

PbO 

1.4298 
.6833 
.7360 

.15527 
.83  458 
.86  688 

PbS04 

PbCr04 
PbCr04 

PbS 

Pb 

PbO 

.7890 
.6411 
.6906 

.89  706 
.80  692 
.83  922 

GRAVIMETRIC  FACTORS 


Given. 

Sought. 

Multiply 

N. 

by  Factor. 
Log. 

PbS 
PbS 

PbCl2 

Pb 

PbO 

Pb 

.8660 
.9329 
.7450 

.93  752 
.96  982 
.87216 

Lithium,  6.94 

PbO 

Pb 

Li2S04 

Pb 
PbO 

Li2 

.9283 

1.0772 

.1263 

.96  770 
.03  230 
.10  123 

Li2S04 
Li3P04 

Li2 

U2O 

Lia 

LiaO 

.2718 
.1797 
2.153 

.43  422 

•25  455 
.33  299 

Magnesium,  24.32 

MgsPzOT 
Mg2P207 
Mg2P207 

Mg2 

(2)MgO 
(2)MgC03 

.2184 
.3621 
.7572 

.33  923 
.55  879 
.87  923 

MgS04 

MgS04 
MgO 

Mg 

MgO 

Mg 

.2020 
.3349 
.6032 

.30  541 
.52  497 
.78  044 

Manganese,  54.93 

MgO 

Mg 

Mn2P207 

MgC03 

MgO 

Mn2 

2.0914 

1.6579 

.3869 

.32  044 
.21  956 
.58  761 

Mn2P207 

Mn304 

Mn304 

(2)MnO 

(3)Mn 

(3)MnO 

.4996 
.7203 
.9301 

.69  863 
.85  749 
.96851 

MnS 
MnS 
MnS04 

Mn 

MnO 

Mn 

.6315 
.8154 
.3638 

.80  034 
.91  136 
.56  086 

MnS04 

MnO 

Mn 

MnO 

Mn02 

MnO 

.4698 
1.2256 
1.2913 

.67  188 
.08  834 
.11  102 

Mercury,  200.6 

Mn 

HgS 

HgS 

MnOa 

Hg 

HgO 

1.5826 
.8622 
.9310 

.19936 
.93  561 
.96  894 

HgCl 
HgCl 
Hg 

Hg 

HgO 

HgO 

.8498 

.9176 

1.0798 

.92  931 
.96  264 
.03  333 

Molybdenum,  96.0 
Nickel,  58.68 

M0O3 

PbMoO* 

NiS04 

Mo 
M0O3 

Ni 

.6667 
.3922 
.3792 

.82  391 
.59  346 
.57  889 

8 


GRAVIMETRIC  FACTORS 


Given. 

Sought. 

Multiply 

N. 

by  Factor. 
Log. 

Nitrogen,  14.01 

NiO 

Ni 

NH4CI 

Ni 
NiO 

N 

.7858 

1.2727 

.2619 

.89  529 
.10471 
.41  807 

NH4CI 
NH4CI 

(NH4)2PtCl6 

NH3 
NH4 

(2)N 

.3184 
.3372 

.06310 

.50  294 
.52  792 
.80  005 

(NH4)2PtCl6 
(NH4)2PtCl6 
(NH4)2PtCl6 

(2)NH3 
(2)NH4 
(2)NH4C1 

.07672 
.08126 
.2410 

.88  492 
.90  990 
.38  198 

Pt 
Pt 
Pt 

(2)N 

(2)NH3 

(2)NH4 

.1435 
.1745 
.1849 

.15699 
.24  186 
.26  684 

N 

NH3 

N 

NHa 

N 
(i)(NH4)20 

1.216 

.8225 
1.859 

.08  487 
.91  513 
.26923 

N 
N 

N 

(i)(NH4)2S04 
(i)N205 

N03 

4.716 
3.855 
4.426 

.67357 
.58  603 
.64  602 

Phosphorus,  31.04 

N 

N 

MgcP207 

N02 

NO 
P2 

3.284 
2.142 

.2787 

.51  641 
.33  083 
.44  519 

Mg2P207 
Mg2P207 

FeP04 

P2O5 

(2)P04 

(|)P206 

.6379 
.8534 
.4708 

.80  477 
.93  117 
.67  287 

U2P20n 

P2O5 

P2 

P2O6 

P2 

P2O6 

.1989 
.4369 
2.289 

.29  854 
.64  042 
.35  958 

Platinum,  195.2 
Potassium,  39.10 

(NH4)2PtCl6 

KCl 
KCl 

Pt 

K 

(4)K20 

.4396 

.52441 

.63170 

.64  306 
.71  967 
.80051 

K2SO4 
K2SO4 
KaPtCle 

K2 

K2O 

K2 

.44876 
.54056 
.1609 

.65  201 
.73  285 
.20  643 

KsPtCle 

K2PtCl6 

KCIO4 

K2O 

(2)KC1 
K 

.1938 
.3067 
.2822 

.28727 
.48  676 

.45  054 

GRAVIMETRIC  FACTORS 


Given. 

Sought. 

Multiply  by 

N. 

'  Factor. 
Log. 

KCIO4 
KCIO4 
K 

(«K20 

KCl 

(i)K20 

.3399 

.5381 

1.2046 

.53  138 
.73  087 
.08  084 

Rubidium,  85.45 

Rb2S04 

Rb2PtCU 

Rb 

Rb2 

Rb2 

Q)Rb20 

.6402 

.2952 

1.093G 

,80  629 

.47017 
.03  887 

Selenium,  79.2 
Silicon,  28.3 

Se 
Se 
Si02 

Se02 
SeOs 
Si 

1.4040 

1.6060 

.4693 

.14737 
.20  576 

.67  147 

SiOa 

Si02 
Si02 

SiOs 

(4)Si207 
Si04 

1.265 
1.398 
1.531 

.10  220 

.14  551 
.18  488 

Silver,  107.88 

Si 

AgCl 

AgCl 

Si02 

Ag 

(4)Ag20 

2.131 
.75262 
.80842 

.32  853 
.87  657 
.90  764 

AgBr 

Agl 

Afe 

Ag 
Ag 

Ag20 

.57444 
.45945 
1.0742 

.75  924 

.66  224 
.03  107 

Sodium,  23.00 

NaCl 
NaCl 
Na2S04 

Na 
(4)Na20 

.39344 
.53028 
.32381 

.59  487 
.72  450 
.51  029 

Na2S04 
NasCOs 
Na^COa 

Na20 

Na^ 

Na20 

.43644 
.43394 

.58487 

.63  992 
.63  743 
.76  706 

Strontium,  87.63 

Na 

SrS04 

SrS04 

(i)Na20 

Sr 

SrO 

1.3478 
.4770 
.5642 

.12  963 
.67  856 

.75  140 

SrCOs 
SrCOs 
Sr 

Sr 

SrO 

SrO 

.5936 

.7019 

1.1826 

.77  346 
.84  630 
.07  284 

Sulphur,  32.06 

BaS04 
BaS04 
BaS04 

S 

SO2 

SO3 

. 13734 
.27443 
.34298 

.13  780 
.43  843 
.53  526 

BaS04 
BaS04 

S 

SO4 

H2SO4 

SO2 

.41151 
.42015 
1.9981 

.61  438 
.62  340 
.30  063 

10 


GRAVIMETRIC  FACTORS 


Given. 

Sought. 

Multiply  by  Factor. 
N.                       Log. 

Tellurium,  127.5 

s 

s 

Te 

SO3 

H2SO4 

TeOa 

2.4972 
3.0591 
1.2510 

.39  746 
.48  560 
.09  725 

Thallium,  204.0 

Te 
Til 
TlzPtCle 

TeOa 

Tl 

TI2 

1.3765 
.6165 
.5000 

.13877 
.78  990 
.69  899 

Thorium,  232.4 
Tin,  1 18.7 

TI2 

Th02 

SnOa 

TI2O 

Th 

Sn 

1.0392 

.8790 
.7877 

.01  671 
.94  398 
.89  634 

Titanium,  48.1 
Tungsten,  184.0 

Sn 

Ti02 

WO3 

SnOz 

Ti 

W 

1.2696 
.6005 
.7931 

.10  366 
.77  852 
.89  933 

Uranium,  238.2 
Vanadium,  51.0 

U3O8 
U3O8 
V2O6 

U3 

(3)U02 
V2 

.8481 
.9620 
.5604 

.92844 
.98319 
.74  853 

Zinc,  65.37 

V2 
V 
ZnO 

V2O5 
VO4 

Zn 

1.784 
2.255 
.8034 

.25  147 
.35313 
.90  492 

ZnS 
ZnS 
ZhzPqOt 

Zn 
ZnO 

Zn2 

.6710 
.8352 
.4289 

.82  669 
.92  177 
.63  237 

Zirconium,  90.6 

Zn2P207 

Zn 
Zr02 

(2)ZnO 

ZnO 

Zr 

.5339 

1.2447 
.7390 

.72  745 
.09  508 
.86  864 

FORMULA  =  WEIGHTS 


11 


Formula. 

Weight. 

Ag 

107.88 

Ag2 

215.76 

Ag3 

323.64 

Ag3As04 

462.60 

AgBr 

187.80 

AgCN 

133.895 

AgCl 

143.34 

Agl 

234.80 

AgI03 

282.80 

AgNOs 

169.89 

Ag20 

231.76 

iAg20 

115.88 

Ag3P04 

418.68 

Ag^S 

247.82 

Al 

27.1 

AI2 

54.2 

AICI3 

133.48 

A1C13-6H20 

241.58 

A1K(S04)2-12H20 

474.51 

A1NH4(S04)2-12H20 

453.45 

AlNa3F6 

210.10 

AI2O3 

102.2 

iAl203 

51.1 

AIPO4 

122.14 

Al2(S04)3 

342.38 

Al2(S04)3-18H20 

666.67 

As 

74.96 

As2 

149.92 

ASCI3 

181.34 

AS2O3 

197.92 

iAS203 

98.96 

iAs203 

49.48 

AS205 

229.92 

iAS206 

114.96 

AS04 

138.96 

AS208 

277.92 

AS2S3 

246.10 

AS2S5 

310.22 

Au 

197.2 

AuCl3-2H20 

339.61 

Table  III. 
FORMULA  =  WEIGHTS 

Log. 
.03  294 
.33  397 
.51  007 
.66  521 

.27370 
.12  676 

.15637 
.37  070 
.45  148 
.23  017 

.36  504 
.06  401 
.62  188 
.39414 

.43  297 
.73  400 
.12  542 
.38306 
.67  625 

.65  653 
•32  243 
.00  945 
.70  842 
.08  686 

•53451 
.82  391 

.87  483 
.17586 
.25  850 
.29  649 
.99  546 
.69  443 
.36  158 
.06  055 
.14  289 
.44  392 
.39  III 
.49  167 

.29491 
.53  098 


Formula. 

Weight. 

Log. 

B 

11.0 

.04  139 

B2 

22.0 

.34  242 

B2O3 

70.0 

.84510 

Ba 

137.37 

.13  789 

BaCOs 

197.375 

.29  530 

BaCl2 

208.29 

.31867 

BaCl2-2H20 

244.32 

.38  796 

BaCr04 

253.37 

.40  376 

Ba(N03)2 

261.39 

.41  729 

BaO 

153.37 

.18574 

BaOa 

169.37 

.22  884 

Ba02-8H20 

313.50 

.49  624 

Ba(OH)2-8H20 

315.51 

.49  901 

BaS04 

233.43 

.36816 

Be 

9.1 

.95  904 

BeO 

25.1 

.39  967 

Bi 

208.0 

.31  806 

Bi2 

416.0 

.61  909 

Bi203 

464.0 

.66  652 

iBi203 

232.0 

.36  549 

BiOCl 

259.46 

.41  407 

Bi(OH)2N03 

304.03 

.48  291 

Bi2S3 

512.18 

.70  942 

Br 

79.92 

.90  266 

Br2 

159.84 

.20369 

Br3 

239.76 

.37  978 

Br4 

319.68 

.50  472 

Br5 

399.60 

.60  163 

Bre 

479.52 

.68  081 

C 

12.005 

.07  936 

C2 

24.01 

.38  039 

CH3 

15.03 

.17  696 

CH4 

16.04 

.20  520 

C2H2 

26.03 

.41  547 

C2H5 

29.05 

.46315 

CeHe 

78.08 

.89  254 

12 


FORMULA  =  WEIGHTS 


Formula. 

Weight. 

C2H3O2 

59.03 

C4H4O6 

148.05 

CN 

26.015 

CNS 

58.08 

CO 

28.005 

C02 

44.005 

C03 

60.005 

C204 

88.01 

CON2H4 

60.06 

CS2 

76.13 

Ca 

40.07 

CaC2 

64.08 

CaCOa 

100.08 

CaCl2 

110.99 

CaCl2-6H20 

219.09 

CaCW 

126.99 

CaF2 

78.07 

CaO 

56.07 

Ca(0H)2 

74.09 

Ca3(P04)2 

310.29 

CaS04 

136.13 

CaS04-2H20 

172.16 

CaSiOa 

116.37 

Cd 

112.40 

CdCOs 

172.41 

CdCl2-2H20 

219.35 

Cd(N03)24H20 

308.48 

CdO 

128.40 

CdS 

144.46 

CdS04 

208.46 

CdS04-2fH20 

256.50 

Ce 

140.25 

Ce203 

328.50 

CeOa 

172.25 

CI 

35.46 

CI2 

70.92 

CI3 

106.38 

CI4 

141.84 

Clfi 

177.30 

Cle 

212.76 

CIO 

51.46 

Log. 

.77  107 

.17041 

.41  522 

.76  403 

.44  724 

.64  350 

77818 

94  453 

77859 

88156 

60  282 

80672 

00  035 

04  528 

34062 

10377 

89248 

74873 

86976 

49  177 

13396 

23593 

06  584 

05077 

23657 

34  114 

48922 

10857 

15975 

31  903 

40909 

14  690 

51654 

23  616 

54  974 

85077 

02  686 

15  180 

24871 

32789 

71  147 

Formula. 

Weight. 

Log. 

CIO3 

83.46 

.92  148 

CIO4 

99.46 

•99  765 

CI2O6 

150.92 

.17875 

CI2O7 

182.92 

.26  226 

Co 

58.97 

.77  063 

CoCl2-6H20 

237.99 

•37  656 

Co(N03)2-6H20 

291.09 

.46  402 

CoO 

74.97 

.87  489 

C03O4 

240.91 

.38  186 

C0SO4 

155.03 

.19  041 

CoS04-7H20 

281 . 14 

.44  892 

Cr 

52.0 

.71  600 

Cr2 

104.0 

.01  703 

Cr203 

152.0 

.18  184 

CrOa 

100.0 

.00  000 

Cr04 

116.0 

.06  446 

CrjOr 

216.0 

.33  445 

Cs 

132.81 

.12323 

CS2 

265.62 

.42  426 

CsAl(S04)2-12H20 

568.22 

•75452 

CsCl 

168.27 

.22  601 

CS2O 

281.62 

.44  966 

CszPtCle 

673.58 

.82  839 

CS2SO4 

361.68 

.55  833 

Cu 

63.57 

.80  325 

CU2 

127  14 

.10428 

CuCl 

99.03 

.99  577 

CUCI2 

134.49 

.12  869 

CuCl2-2H20 

170.52 

.23  177 

Cul 

190.49 

.27  987 

CuFeSa 

183.53 

.26371 

Cu(N03)2-6H20 

295.69 

.47  083 

CuO 

79.57 

•90  075 

CU2O 

143.14 

.15576 

CU2S 

159.20 

.20  194 

CuSCN 

121.65 

.08  511 

CUSO4 

159.63 

.20311 

CuS04-5H20 

249.71 

.39  744 

Fe 

55.84 

.74  695 

Fe2 

111.68 

.04  798 

FORMULA  =  WEIGHTS 


13 


Formula. 

Weight. 

Log. 

Formula. 

Weight. 

Log. 

Fe3 

167.52 

.22  406 

Hg 

200.6 

.30  233 

FeCOs 

115.85 

.06  390 

Hg(CN)2 

252.63 

.40  248 

FeCls 

162.22 

.21  010 

HgCl 

236.06 

.37  302 

FeCl3-6H20 

270.32 

.43  188 

HgCl2 

271.52 

.43  380 

Fe(NH4)2(S04)2-6H20  392 .  14 

.59  344 

Hgl2 

454.44 

.65  748 

FeNH4(S04)2-12H20 

482.19 

.68322 

HgO 

216.6 

.33  566 

FeO 

71.84 

.85637 

HgS 

232.66 

.36  672 

FeaOs 

159.68 

.20325 

iFesOa 

79.84 

.90  222 

I 

126.92 

.10353 

Fe304 

231.52 

.36  459 

I2 

253.84 

.40  456 

FeS 

87.90 

.94  399 

I3 

380.76 

.58  065 

FeS2 

119.96 

.07  904 

I4 

507.68 

.70  559 

FeS04-7H20 

278.01 

.44  406 

I5 

634.60 

.80  250 

le 

761.52 

.88  168 

H 

1.008 

.00  346 

I2O5 

333.84 

.52  354 

H2 

2.016 

.30  449 

IO3 

174.92 

.24  284 

H3 

3.024 

.48  058 

IO4 

190.92 

.28  085 

H4 

4.032 

.60  552 

H5 

5.040 

.70  243 

K 

39.10 

.59218 

He 

6.048 

.78  161 

K2 

78.20 

.89321 

H3BO3 

62.02 

.79  253 

K3 

117.30 

.06  930 

HBr 

80.93 

.90  811 

K4 

156.40 

.19424 

HC2H3O2 

60.04 

.77  844 

KA1(S04)2-12H20 

474.51 

.67  625 

HCN 

27.023 

.43  174 

KBr 

119.02 

.07  562 

HCO2H 

46.02 

.66  295 

KBrOs 

167.02 

.22  277 

H2C2O4 

90.03 

.95  439 

KCN 

65.12 

.81371 

H2C204-2H20 

126.06 

.10  058 

K2CO3 

138.21 

.14054 

i[H2C204-2H201 

63.03 

.79  955 

KCl 

74.56 

.87251 

H2C'4H406 

150.07 

.17  629 

2KC1 

149.12 

.17354 

HCl 

36.468 

.56  192 

KCIO3 

122.56 

.08  835 

HCIO4 

100.47 

.00  204 

KCIO4 

138.56 

.14  164 

HF 

20.01 

.30125 

K2Cr04 

194.20 

.28825 

HI 

127.93 

.10  697 

K2Cr207 

294.20 

.46  864 

HKCO3 

100.11 

.00  048 

iK2Cr07 

49.033 

.69  049 

HNO3 

63.02 

.79  948 

KCr  (804)2- I2H2O 

499.41 

.69  846 

HNaC03 

84.01 

.92  433 

K3Fe(CN)6 

329.23 

.51  750 

HNa2P04l2H20 

358.24 

.55418 

K4Fe(CN)6 

368.33 

.56  624 

HO 

17.008 

.23  065 

K4Fe(CN)6-3H20 

422.38 

.62  570 

H2O 

18.016 

.25  565 

KHCO3 

100.11 

.00  048 

H2O2 

34.02 

.53  173 

KI 

166.02 

.22  016 

H3PO4 

98.06 

.99  149 

KIO3 

214.02 

.33  045 

H2S 

34.08 

.53  250 

iKlOs 

35.67 

.55230 

H2SO4 

98.08 

.99  158 

KMn04 

158.03 

.19874 

iH2S04 

49.04 

•69  055 

2KMn04 

316.06 

.49  977 

H2SiF6 

144.32 

.15933 

iKMn04 

31.606 

.49  977 

14 


FORMULA  =  WEIGHTS 


Formula. 

Weight. 

Log. 

Formula. 

Weight. 

Log. 

KNO2 

85.11 

.92  998 

Mo 

96.0 

.98  227 

KNO3 

101.11 

.00  479 

M0O3 

144.0 

.15836 

KNaC4H4064H20 

282.22 

.45  059 

MoS, 

160.1 

.20  439 

K2O 

94.20 

.97  405 

iKaO 

47.10 

.67  302 

N 

14.01 

.14644 

KOH 

56.11 

.74  904 

N2 

28.02 

.44  747 

KsPtCle 

486.16 

.68  678 

NH2 

16.026 

.20  482 

JKsPtCle 

243.08 

•38  575 

NH3 

17.034 

.23  131 

KSCN 

97.18 

.98  758 

NH4 

18.042 

.25  629 

K2SO4 

174.26 

.24  120 

(NH4)2 

36.084 

.55  732 

4K2SO4 

87.13 

•94  017 

NH4A1  (804)2- 12H20 

453.45 

.65  653 

K2SiF6 

220.5 

.34341 

(NH4)2C20rH20 

142.11 

.15  262 

N2H4CO 

60.06 

.77  859 

Li 

6.94 

.84  136 

NH4CI 

53.502 

.72  837 

Liz 

13.88 

.14239 

NH4Fe(S04)2-12H20 

482.19 

.68322 

U2CO3 

73.89 

.86  859 

(NH4)2Fe(S04)2-6H20392.14 

.59  344 

LiCl 

42.40 

.62  737 

NH4NaHP044H20 

209.15 

.32  046 

Li20 

29.88 

.47  538 

(NH,)20 

52.084 

.71  670 

iLi20 

14.94 

.17435 

NH4OH 

35.05 

.54  469 

Li3P04 

115.86 

.06  393 

NH4NO3 

80.05 

.90  336 

Li2S04 

109.94 

.04  116 

(NH4)2PtCl6 

444.04 

.64  742 

Mg 

24.32 

.38  596 

NH4SCN 

76.12 

.88  150 

Mg2AS207 

310.56 

.49214 

(NH4)2S04 

132.144 

.12  104 

MgCOs 

84.325 

.92  596 

N20 

44.02 

.64  365 

MgCl2-6H20 

203.34 

.30  822 

NO 

30.01 

•47  727 

MgO 

40.32 

.60  552 

N02 

46.01 

.66  285 

Mg2P207 

222.72 

.34  776 

N03 

62.01 

.79  246 

MgS04 

120.38 

.08  055 

N203 

76.02 

.88  093 

MgS04-7H20 

246.49 

.39  180 

N205 

108.02 

.03  350 

MgSiOa 

100.62 

.00  268 

iN205 

54.01 

.73  247 

Mn 

54.93 

.73  981 

Na 

23.00 

.36  173 

Mn2 

109.86 

.04  084 

Na2 

46.00 

.66  276 

MnCOs 

114.94 

.06  047 

NagAlFe 

210.1 

•32  243 

MnCl24H20 

197.91 

.29  647 

NasB.Oy 

202.0 

•30  535 

MnO 

70.93 

.85  083 

Na2B4O7l0H2O 

382.16 

.58  225 

MnzOa 

157.86 

.19828 

NaBr 

102.92 

.01  250 

Mn304 

228.79 

.35  944 

NaC2H302 

82.03 

.91  397 

Mn02 

86.93 

.93917 

NaC2H302-3H20 

136.08 

•13379 

Mn207 

221.86 

.34  608 

Na2C03 

106.005 

•02  533 

Mn2P207 

283.94 

.45  323 

NaaCOglOHaO 

286.17 

.45  662 

MnS 

86.99 

.93  947 

NaHCOa 

84.01 

•92  433 

MnS04 

150.99 

.17895 

NaCl 

58.46 

.76  686 

MnS044H20 

223.05 

.34  840 

NaC103 

106.46 

.02  719 

MnS04-7H20 

277.10 

.44  264 

NaC104 

122.46 

.08  799 

FORMULA  =  WEIGHTS 


15 


Formula. 

Weight. 

Log. 

Formula. 

Weight. 

Log. 

NaF 

42.0 

.62  325 

Pb 

207.20 

.31  639 

Na2HP04l2H20 

358.24 

.55418 

Pb(C2H302)2-3H20 

379.32 

.57  900 

NaHSOs 

104.07 

.01  733 

PbC03 

267.21 

.42  686 

NaHS04 

120.07 

.07  943 

PbCl2 

278.12 

.44423 

Nal 

149.92 

.17586 

PbCr04 

323.20 

.50  947 

NaNOz 

69.01 

.83  891 

Pb(N03)2 

331.22 

.52  012 

NaNOs 

85.01 

.92  947 

PbO 

223.20 

.34  869 

NasO 

62.00 

.79  239 

Pb02 

239.20 

.37  876 

iNazO 

31.00 

.49  136 

Pb304 

685.60 

.83  607 

Na202 

78.00 

.89  209 

PbS 

239.26 

.37  887 

NaOH 

40.01 

.60  217 

PbS04 

303.26 

.48  181 

NaPOa 
Na4P207 

NaaS 

102.04 

266.08 

78.06 

.00  877 
.42  501 
.89  243 

Pd 

Pdl2 

106.7 
360.54 

.02  816 
.55  696 

Na2S04 

Na2SO4-10H2O 

Na2S203-5H20 

142.06 
322.22 
248.20 

.15247 
.50816 
.39  480 

Pt 

PtCLt 
PtCle 

195.2 

337.04 

407.96 

.29  048 
.52  768 
.61  062 

Ni 

NiCl2-6H20 

Ni(N03)2-6H20 

NiO 

NiS 

NiS04 

NiS04-6H20 

58.68 
237.70 
290.80 
74.68 
90.74 
154.74 
262.84 

.76  849 
.37  603 
.46  359 
.87  320 
.95  780 
.18  960 
.41  969 

Rb 

RbAl(S04)2-12H20 

RbCl 

Rb20 

RbaPtCle 

Rb2S04 

85.45 
520.86 
120.91 
186.90 
578.86 
266.96 

.93  171 
.71  672 
.08  246 
.27  161 
.76257 
.42  645 

NiS04-7H20                 280.85 
Ni(NH4)2(S04)2-6H20  394 .  98 

.44  848 
.59  658 

S 

S2 

32.06 
64.12 

.50  596 
.80  699 

0 
O2 
O3 
O4 

16.00 
32.00 
48.00 
64.00 

.20  412 

.50515 
.68  124 
.80618 

SON 
SO2 
SO3 
SO4 

58.075 
64.06 
80.06 
96.06 

.76  399 
.80  659 
.90  342 
.98  254 

Os 
Oe 
OH 

80.00 
96.00 
17.008 

.90  309 
.98  227 
.23  065 

Sb 

2Sb 

SbCla 

120.2 
240.4 
226.58 

.07  990 
.38  093 
.35  522 

P 

31.04 

.49  192 

SbCIs 

297.50 

.47  349 

P2 

62.08 

.79  295 

SbaOa 

288.4 

.46  000 

PCI3 

137.42 

.13805 

iSb203 

144.2 

.15897 

PCI5 

208.34 

.31  877 

Sb205 

320.4 

.50  569 

P2O5 

142.08 

.15253 

iSb205 

160.2 

.20  466 

^P205 

71.04 

.85  150 

Sb204 

304.4 

.48  344 

PO4 

95.04 

.97  791 

SbOCl 

171.7 

.23  477 

2PO4 

190.08 

.27  894 

SbOKC4H406-iH20 

332.36 

.52  161 

P2O3 

110.08 

.04  171 

Sb2S3 

336.6 

.52  7" 

P2O7 

174.08 

.24  075 

SbaSs 

400.7 

.60  282 

16 


FORMULA  =  WEIGHTS 


Formula. 

Weight. 

Log. 

Formula. 

Weight. 

Log. 

Se 

79.2 

.89  873 

Th 

232.4 

.36  624 

Se02 

111.2 

.04  610 

ThOa 

264.4 

.42  226 

Se03 

127.2 

.10449 

Ti 

48.1 

.68215 

Si 

28.3 

.45  179 

TiOa 

80.1 

•90  363 

Si2 

SiF4 
SiF6 

Si02 

SiOa 
Si04 

56.6 
104.3 
142.3 
60.3 
76.3 
92.3 

.75  282 
.01  828 

.15320 
.78  032 

.88  252 
.96  520 

Tl 

TlCl 
Til 
TINO3 

Tl2PtCl6 

204.0 

239.46 

330.92 

266.01 

815.96 

.30  963 
.37  923 
•51  973 
.42  490 
.91  167 

SisOs 

212.9 

.32818 

U 

238.2 

.37  694 

Sn 

SnCl2 

SnCl2-2H20 

SnCU 

SnO 

118.7 

189.62 

225.65 

260.54 

134.7 

.07  445 
.27  789 

.35  344 
.41  587 
.12937 

UO2 
U3O8 

U02(C2H302)2-2H20 

U02(N03)2-6H20 

U2P20U 

270.2 
842.6 
424.30 
502.32 

714.48 

.43  169 
.92  562 
.62  767 
.70  098 
.85  399 

Sn02 

SnS 

SnSg 

150.7 
150.76 

182.82 

.17811 
.17828 
.26  203 

V 

V205 

51.0 
182.0 

.70  757 
.26  007 

Sr 
SrCOs 

87.63 
147.64 

.94  265 
.16  920 

w 
W03 

184.0 
232.0 

.26  482 
.36  549 

SrCl2-6H20 

266.65 

.42  594 

Zn 

65.37 

.81  538 

Sr(N03)2 

211.65 

.32  562 

ZnCOs 

125.38 

.09  823 

SrO 

103.63 

.01  549 

ZnCl2 

136.29 

.13446 

Sr(OH)2-8H20 

265.77 

.42  450 

ZnO 

81.37 

.91  046 

SrS04 

183.69 

.26  409 

ZnS 

97.43 

.98  869 

ZnS04 

161.43 

.20  798 

Te 

Te02 

127.5 
159.5 

.10551 
.20  276 

ZnS04-7H20 

287.54 

.45  870 

TeOs 

175.5 

.24  428 

Zr 

90.6 

.95713 

TeOrSHaO 

229.55 
18.016 

.36  088 

Zr02 

122.6 
234.21 

.08  849 

H2O 

.25  565 

I3H2O 

.36  961 

2H2O 

36.032 

.55  668 

I4H2O 

252.22 

.40  178 

3H2O 

54.048 

.73  278 

I5H2O 

270.24 

.43  175 

4H2O 

72.064 

.85  771 

I6H2O 

288.26 

.45  978 

5H2O 

90.08 

.95  463 

I7H2O 

306.27 

.48  611 

6H2O 

108.10 

.03  383 

I8H2O 

324.29 

.51  094 

7H2O 

126.11 

.10  075 

I9H2O 

342.30 

.53  441 

8H2O 

144.13 

.15  875 

2OH2O 

360.32 

.55  668 

9H2O 

162.14 

.20  989 

2IH2O 

378.34 

.57  788 

IOH2O 

180.16 

.25  565 

22H2O 

396.35 

.59  808 

IIH2O 

198.18 

.29  706 

23H2O 

414.37 

.61  739 

I2H2O 

216.19 

.33  484 

24H2O 

432.38 

.63  587 

INDIRECT  ANALYSES 


17 


NazSO 
K2SO. 


:•! 


Na2S04l 
Li2S04  ) 


RbaPtCle 
CszPtCle 


Table  IV. 
INDIRECT  ANALYSES. 


Mixture. 
a. 

Determined. 
6. 

AgBr 
Agl 

1 

Loss  by  fusing 
in  chlorine 

AgBr 
Agl 

1 

Loss  by  fusing 
in  chlorine. 

NaCl 
KCl 

CI 

BaS04 


BaS04 


RbsSO. 

CsaSO. 


:'l 


Formula. 


AgBr  =  2.5496  a  -  6.5454  b 

(log  .40  647)     (log  .81  594) 
Agl=  -  1.5496  a  +  6.5454  fc 
(log  .19  022)     (log  .81  594) 

Br  =  1.0850  a-  2.7855  6 

(log  .03  543)     (log  .44  490) 
I  =  -  .83758  a  +  3.5380  b 
(log  .92  303)     (log  .54  876) 

NaCl  =  -  3.6310  a  +  7.6348  b 

(^log  .56  003)     (log  .88  280) 
KCl  =  4.6310  a -7.6348  6 
(log  .66  567)     (log  .88  280) 

Na2S04  =  -  4.4108  a  +  3.2928  b 
(log  .64  452)     (log  .51  756) 
K2S04  =  5.4108  a -3.2928  6 

(log  .73  326)     (log  .51  756) 

Na2S04  =  4.4228  a  -  2.0830  b 

(log  .64  570)     (log  .31  869) 
Li2S04  =  -  3.4228  a  +  2.0830  b 
(log  .53438)     (log  .31  869) 

Rb2S04  =  3.2682  a  -  6.0866  b 

(log  .51  431)     (log  .78  437) 

CS2SO4  =  -  3.2682  a  +  7.0866  b 
(log  .51  431)     (log  .85  044) 


Note.  —  This  table  is  given  to  show  a  few  convenient  formulas  for  calculating  the  results  of 
indirect  analyses  in  cases  where  two  transformations  occur  together.  The  formulas  themselves 
may  be  readily  derived  from  the  proper  factors  by  the  use  of  algebra.  For  instance,  for  the 
third  pair  of  formulas  given  above  we  have: 

NaCl  +  KCl  =  a,  and  .60656  NaCl  +  .47559  KCl  =  b, 

and  the  solution  of  these  simultaneous  equations  gives  the  formulas. 

Of  the  formulas  given  in  the  table,  the  first  two  pairs  represent  a  process  that  has  been  fre- 
quently employed,  and  is  capable  of  giving  good  results.  The  student  should  be  warned,  how- 
ever, against  the  use  of  most  of  the  methods  of  complex  indirect  determination,  except  for  the 
purpose  of  obtaining  approximate  results,  because  the  errors  in  the  determinations  are  largely 
increased  in  the  final  results.  For  example,  the  formulas  indicate  that  an  error  of  one  part  in 
100  in  the  determination  of  chlorine  in  a  mixture  of  NaCl  and  KCl  would  produce  errors  of  7.6 
parts  in  100  in  the  calculated  amounts  of  the  salts. 


18 


REDUCTION   OF  GAS-VOLUMES 


TABLE  V. 


REDUCTION  OF  GAS-VOLUMES  TO  0°  AND  760  MM. 
Vol.  at  0°  and  760  mm  =  v.  F,  (h—w). 


^"  &"^  \^^^  f 

J.-   —  M.  <^\^\j\ji     iiyj 

--^        760(1+. 003670' 

<=  Temperature  of  gas 

(C.°); 

/i=  Barometer 

-height,  corrected, 

(mm); 

w.'=  Tension  of  aqueous  vapor  (mm). 

•    t°. 

Log  F* 

w. 

t°. 

Log  F* 

w. 

0 

.11  919 

4.6 

15 

.09592 

12.7 

0.5 

.11839 

4.8 

15.5 

•09517 

13.1 

.11  760 

4.9 

16 

.09441 

13.5 

1.5 

.11  681 

5.1 

16.5 

.09366 

13.9 

2 

.11  602 

5.3 

17 

.09291 

14.4 

2.5 

.11  523 

5.5 

17.5 

.09  216 

14.9 

3 

.11  444 

5.7 

18 

.09  141 

15.3 

3.5 

.11365 

5.9 

18.5 

.09  067 

15.8 

4 

.11  286 

6.1 

19 

.08  992 

16.3 

4.5 

.11  208 

6.3 

19.5 

.08918 

16.8 

5 

II  129 

6.5 

20 

.08843 

17.4 

5.5 

.11  051 

6.7 

20.5 

.08  769 

17.9 

6 

•10973 

7.0 

21 

.08  695 

18.5 

6.5 

10  895 

7.2 

21.5 

.08  621 

19.0 

7 

10  817 

7.5 

22 

.08  547 

19.6 

7.5 

10  740 

7.7 

22.5 

.08474 

20.2 

8 

10  662 

8.0 

23 

.08  400 

20.9 

8.5 

10585 

8.3 

23.5 

.08327 

21.5 

9 

10  508 

8.5 

24 

•08  253 

22.2 

9.5 

10430 

8.8 

24.5 

.08  180 

22.8 

10 

10354 

9.1 

25 

.08  107 

23.5 

10.5 

10277 

9.4 

25.5 

.08034 

24.2 

11 

10  201 

9.8 

26 

.07  961 

25.0 

11.5 

10  124 

10.1 

26.5 

.07  888 

25.7 

12 

10  048 

10.4 

27 

.07  816 

26.5 

12.5 

09972 

10.8 

27.5 

•07  743 

27.3 

13 

09  896 

11.1 

28 

.07  671 

28.1 

13.5 

09  820 

11.5 

28.5 

.07  599 

28.9 

14 

09  744 

11.9 

29 

.07  527 

29.7 

14.5 

09  668 

12.3 

29.5 

•07455 

30.6 

30 

•07  383 

31.5 

*  The  characteristic  of  all  the  logarithms  is  —3. 


CALCULATION  OF  PERCENTAGE  OF  NITROGEN 


19 


Table  VI. 

CALCULATION  OF  PERCENTAGE  OF  NITROGEN  FROM  THE 

GAS-VOLUME. 


%N  = 


v.F,  {h  —  w) 


Volume  of  nitrogen  (cc) . 

.0012510  X  100 


aOO  g.  H2O  +  30  g.  KOH  =  23.1%  KOH) 


^  =  Factor  (table)  =^gQ^^^^Q3g^^^ 

h  =  Height  of  barometer,  corrected  (mm). 
w  =  Tension  of  aqueous  vapor  (mm). 
S  =  Weight  of  substance  (g) . 
t  =  Temperature  of  gas  (C.°). 


23.1%  KOH 

H2O. 

23.1%  KOH. 

H2O. 

r. 

Log.  F* 

w. 

w. 

r.             ] 

Log  F.* 

w. 

w. 

0 

.21  645 

3.7 

4.6 

15.5 

19243 

10.5 

13.1 

0.5 

•21  565 

3.8 

4.8 

16 

19  167 

10.8 

13.5 

1 

.21  486 

3.9 

4.9 

16.5 

19  092 

11.2 

13.9 

1.5 

.21  407 

4.1 

5.1 

17 

19  017 

11.5 

14.4 

2 

.21  328 

4.2 

5.3 

17.5 

18  942 

11.9 

14.9 

2.5 

.21  249 

4.4 

5.5 

18 

18867 

12.3 

15.3 

3 

.21  170 

4.5 

5.7 

18.5 

18793 

12.7 

15.8 

3.5 

.21  091 

4.7 

5.9 

19 

18  718 

13.1 

16.3 

4 

.21  012 

4.9 

6.1 

19.5 

18  644 

13.5 

16.8 

4.5 

.  20  934 

5.0 

6.3 

20 

18569 

13.9 

17.4 

5 

.20855 

5.2 

6.5 

20.5 

18495 

14.4 

17.9 

5.5 

.20777 

5.4 

6.7 

21 

18  421 

14.8 

18.5 

6 

.  20  699 

5.6 

7.0 

21.5 

18347 

15.3 

19.0 

6.5 

.20  621 

5.8 

7.2 

22 

18273 

15.8 

19.6 

7 

•  20  543 

6.0 

7.5 

22.5 

18  200 

16.3 

20.2 

7.5 

.  20  466 

6.2 

7.7 

23 

18  126 

16.8 

20.9 

8 

.20388 

6.4 

8.0 

23.5 

18053 

17.3 

21.5 

8.5 

.20311 

6.6 

8.3 

24 

17979 

17.8 

22.2 

9 

.20  234 

6.8 

8.5 

24.5 

17906 

18.4 

22.8 

9.5 

.20  156 

7.0 

8.8 

25 

17833 

18.9 

23.5 

10 

.  20  080 

7.3 

9.1 

25.5 

17  760 

19.5 

24.2 

10.5 

.  20  003 

7.6 

9.4 

26 

17687 

20.1 

25.0 

11 

.19927 

7.8 

9.8 

26.5 

17  614 

20.7 

25.7 

11.5 

.19  850 

8.1 

10.1 

27 

17  542 

21.3 

26.5 

12 

.19774 

8.3 

10.4 

27.5 

17469 

21.9 

27.3 

12.5 

.19  698 

8.6 

10.8 

28 

17397 

22.6 

28.1 

13 

.19  622 

8.9 

11.1 

28.5 

17325 

23.3 

28.9 

13.5 

.19546 

9.2 

11.5 

29 

17  253 

24.0 

29.7 

14 

.19470 

9.5 

11.9 

29.5 

17  181 

24.7 

30.6 

14.5 

19394 

9.8 

12.3 

30 

17  109 

25.4 

31.5 

15 

.19318 

10.1 

12.7 

Note. 

—  The  table  is 

based  on 

Lord  Raylei 

?h's  weight  of  n 

itrogen,  at  latitude  41* 

'  and  sea- 

level,  0.0012510  g. 

The  characteristic  of  all  the  logarithms  is  —  4. 


20 


BAROMETER   CORRECTIONS 


Table  VII. 

BAROMETER  CORRECTIONS. 

For  Temperature. 


Grlass  Scale  (Bunsen). 

Brass  Scale  (Delcros). 

mm.  to  be  deducted. 

mm.  to  be  deducted. 

^o 

Reading. 

f. 

Reading. 

{  . 

700 

720 

740 

760 

780 

700 

720 

740 

760 

780 

1 

0.120 

0.123 

0.127 

0.130 

0.133 

1 

0.113 

0.116 

0.119 

0.123 

0.126 

2 

0.2 

0.2 

0.3 

0.3 

0.3 

2 

0.2 

0.2 

0.2 

0.3 

0.3 

3 

0.4 

0.4 

0.4 

0.4 

0.4 

3 

0.3 

0.4 

0.4 

0.4 

0.4 

4 

0.5 

0.5 

0.5 

0.5 

0.5 

4 

0.5 

0.5' 

0.5 

0.5 

0.5 

5 

0.6 

0.6 

0.6 

0.7 

0.7 

5 

0.6 

0.6 

0.6 

0.6 

0.6 

6 

0.7 

0.7 

0.8 

0.8 

0.8 

6 

0.7 

0.7 

0.7 

0.7 

0.8 

7 

0.8 

0.9 

0.9 

0.9 

0.9 

7 

0.8 

0.8 

0.8 

0.9 

0.9 

8 

1.0 

1.0 

1.0 

1.0 

1.1 

8 

0.9 

0.9 

1.0 

1.0 

1.0 

9 

1.1 

1.1 

1.1 

1.2 

1.2 

9 

1.0 

1.0 

1.1 

1.1 

1.1 

10 

1.2 

1.2 

1.3 

1.3 

1.3 

10 

1.1 

1.2 

1.2 

1.2 

1.3 

11 

1.3 

1.4 

1.4 

1.4 

1.5 

11 

1.2 

1.3 

1.3 

1.4 

1.4 

12 

1.4 

1.5 

1.5 

1.6 

1.6 

12 

1.4 

1.4 

1.4 

1.5 

1.5 

13 

1.6 

1.6 

1.7 

1.7 

1.7 

IS 

1.5 

1.5 

1.6 

i.e 

1.6 

U 

1.7 

1.7 

1.8 

1.8 

1.9 

H 

1.6 

1.7 

1.7 

1.7 

1.8 

15 

1.8 

1.8 

1.9 

2.0 

2.0 

15 

1.7 

1.8 

1.8 

1.8 

1.9 

16 

1.9 

2.0 

2.0 

2.1 

2.1 

16 

1.8 

1.9 

1.9 

2.0 

2.0 

17 

2.0 

2.1 

2.2 

2.2 

2.3 

17 

1.9 

2.0 

2.0 

2.1 

2A 

18 

2.2 

2.2 

2.3 

2.3 

2.4 

18 

2.0 

2.1 

2.2 

2.2 

2.3 

19 

2.3 

2.3 

2.4 

2.5 

2.5 

19 

2.1 

2.2 

2.3 

2.3 

2.4 

20 

2.4 

2.5 

2.5 

2.6 

2.7 

20 

2.3 

2.3 

2.4 

2.5 

2.5 

21 

2.5 

2.6 

2.7 

2.7 

2.8 

21 

2.4 

2.5 

2.5 

2.6 

2.7 

22 

2.6 

2.7 

2.8 

2.9 

2.9 

22 

2.5 

2.6 

2.6 

2.7 

2.8 

23 

2.8 

2.8 

2.9 

3.0 

3.1 

23 

2.6 

2.7 

2.8 

2.8 

2.9 

H 

2.9 

3.0 

3.0 

3.1 

3.2 

U 

2.7 

2.8 

2.9 

2.9 

3.0 

25 

3.0 

3.1 

3.2 

3.3 

3.3 

25 

2.8 

3.0 

3.1 

3.0 

3.2 

26 

3.1 

3.2 

3.3 

3.4 

3.5 

26 

2.9 

3.1 

3.1 

3.2 

3.3 

27 

3.2 

3.3 

3.4 

3.5 

3.6 

27 

3.1 

3.2 

3.2 

3.3 

3.4 

28 

3.4 

3.4 

3.5 

3.6 

3.7 

28 

3.2 

3.3 

3.3 

3.4 

3.5 

29 

3.5 

3.6 

3.7 

3.8 

3.9 

29 

3.3 

3.4 

3.5 

3.6 

3.7 

30 

3.6 

3.7 

3.8 

3.9 

4.0 

30 

3.4 

3.5 

3.6 

3.7 

3.8 

Notes  on  Densities  of  Gases  and  Vapors.  —  The  molecular  weight  in  grams  of  any  gas  or 
vapor  has  a  volume  of  22.4  liters  at  0°  and  760  mm.     Hence: 

Mol.  wt.  =  22.4  X  wt.  of  1  liter  in  grams, 
and  Wt.  of  1  hter  in  grams.  =  mol.  wt.  -7-  22.4. 

In  most  cases  the  values  derived  from  these  formulas  do  not  correspond  precisely  with  the 
observed  values,  but  they  suffice  for  most  practical  purposes.  For  instance,  the  volume  of  32  g. 
of  oxygen  is  22.390  instead  of  22.4  1.,  and  in  other  cases  there  is  still  greater  variation. 

It  happens  that  the  relation  between  grams  and  liters  is  practically  the  same  as  that  be- 
tween ounces,  avoirdupois  and  cubic  feet,  hence: 

Wt.  of  1  cu.  ft.  in  ounces,  av.  =  mol.  wt.  -^  22.4. 

The  density  of  air  corresponds  to  a  mean  molecular  weight  of  29,  or  more  precisely  28.95 
when  compared  with  O2  as  32.  Hence  the  molecular  weight  of  a  gas  or  vapor  divided  by  29 
gives  its  density  compared  with  air  at  the  same  temperature  and  pressure.  1  1.  air  at  0°  and 
760  mm.  =  1.2928  g. 


MULTIPLES  FOR  ORGANIC  COMPOUNDS 


Table  VIII. 
MULTIPLES  FOR  ORGANIC  COMPOUNDS. 


Number. 

H 

1.008 

Hj 

2.016 

Ha 

3.024 

H4 

4.032 

H5 

5.040 

He 

6.048 

H, 

7.056 

Hs 

8.064 

H, 

9.072 

Hio 

10.08 

Hu 

11.09 

H,2 

12.10 

Hi3 

13.10 

H,4 

14.11 

Hi5 

15.12 

H16 

16.13 

H„ 

17.14 

His 

18.14 

Hi, 

19.15 

H20 

20.16 

H21 

21.17 

H22 

22.18 

H23 

23.18 

H« 

24.19 

H25 

25.20 

H.a 

26.21 

H2, 

27.22 

H28 

28.22 

H29 

29.23 

H30 

30.24 

H31 

31.25 

H32 

32.26 

H33 

33.26 

H34 

34.27 

H35 

35.28 

H36 

36.29 

H3, 

37.30 

Has 

38.30 

H3, 

39.31 

H40 

40.32 

Log. 

Number. 

Log. 

Number. 

Log. 

.00  346 

C 

12.005 

.07  936 

N 

14.01 

.14644 

.30  449 

C2 

24.01 

.38  039 

N2 

28.02 

.44  747 

.48  058 

C3 

36.02 

.55  654 

N3 

42.03 

.62  356 

.60  552 

C4 

48.02 

.68  142 

N4 

56.04 

.74  850 

.70  243 

C5 

60.03 

.77  837 

N5 

70.05 

.84  541 

.78  161 

Ce 

72.03 

•85751 

Ne 

84.06 

.92  459 

.84  856 

C7 

84.04 

.92  449 

N7 

98.07 

.99  154 

.90  655 

Cs 
C9 

96.04 
108.05 

.98  245 
.03  362 

Ns 

112.08 

.04  953 

.95  770 

S 

32.06 

•50  596 

.00  346 

Cio 

120.05 

.07  936 

S2 

64.12 

.80  699 

.04  493 

Cn 

132.06 

.12  077 

S3 

96.18 

.98  308 

.08  279 

C12 

144.06 

.15854 

S4 

128.24 

.10  802 

.11  727 

Cu 

156.07 

.19332 

S5 

160.30 

.20  493 

.14953 

Ci4 
Cl5 

168.07 
180.08 

.22  549 
.25  546 

So 

192.36 

.28  412 

.17955 

CI 

35.46 

.54974 

.20  763 

C16 

192.08 

.28348 

C12 

70.92 

.85  077 

.23  401 

Ci, 

204.09 

.30  982 

C13 

106.38 

.02  686 

.25  864 

Cis 

216.09 

.33  463 

C14 

141.84 

.15  180 

.28217 

Ci9 

228.10 

.35813 

C15 

177.30 

.24  871 

.30  449 

C20 

C21 

240.10 
252.11 

.38  039 
.40  159 

Ch 

212.76 

.32  789 

.32  572 

Br 

79.92 

.90  266 

.34  596 

C22 

264.11 

.42  179 

Bra 

159.84 

.20  369 

.36511 

Cs, 

276.12 

.44  no 

Bra 

239.76 

.37  978 

.38  364 

C24 

0^5 

288.12 
300.13 

.45  957 
.47731 

Br4 

319.68 

.50  472 

.40  140 

126.92 

.10353 

.41  847 

C26 

312.13 

.49  433 

I2 

253.84 

.40  456 

.43  489 

c« 

324.14 

.51  073 

I3 

380.76 

.58  065 

.45  056 

C2, 

336.14 
348.15 

.52  652 
.54  177 

1 4 

507.68 

.70  559 

.46  583 

On 

176 

.24551 

.48  058 

C30 

360.15 

.55  648 

O12 

Oi3 

192 

208 

.28330 

.49  485 

0 

16 

.20  412 

.31  806 

.50  866 

02 

32 

.50515 

Ou 

224 

.35  025 

.52  192 

03 

48 

.68  124 

Oi5 

240 

.38  021 

.53  491 

04 

64 

.80618 

O16 

256 

.40  824 

.54  753 

05 

80 

.90  309 

Ol7 

272 

.43  457 

.55  979 

Oe 

96 

.98227 

O18 

288 

.45  939 

•57  171 

O7 

112 

.04  922 

Oi9 

304 

.48  287 

.58  320 

Os 

128 

.10  721 

O20 

320 

.50515 

.59  450 

O9 

144 

.15836 

O21 

336 

.52  634 

.60  552 

Oio 

160 

.20  412 

O22 

352 

.54  654 

22     CONSTANTS  FOR  MOLECULAR-WEIGHT  DETERMINATIONS 


TABLE  IX. 
CONSTANTS  FOR  MOLECULAR-WEIGHT  DETERMINATIONS. 

(The  constants  are  taken  from  Ostwald-Luther  "  Physiko-Chemische  Messungen.") 

The  formula  is  M  =  K  -jj,  where  M  is  the  molecular  weight  of  the  dis- 
solved substance,  s  its  weight  in  grams,  L  the  weight  of  the  solvent  in  grams, 
J  the  observed  change  in  temperature,  and  K  is  the  constant,  which  depends 
upon  the  nature  of  the  solvent. 

By  Depression  of  Freezing-point. 

, K. . 

Solvent. 
Water 
Benzole 
Naphthaline 
Phenanthrene 
Phenol 

Glacial  acetic  acid 
Nitrobenzole 
P.  toluidine 


Solvent. 
Ethyl  ether 
Benzole 
Chloroform 
Carbon  bisulphide 
Glacial  acetic  acid 
Ethyl  alcohol 
Ethyl  acetate 
Acetone 
Water 

Ethylene  bromide 
Aniline 
Phenol 


Melts  at. 

N. 

Log. 

0° 

1850 

.2672 

5.5 

5000 

.6990 

80 

7000 

.8451 

96 

12000 

.0792 

39 

7500 

•8751 

17.7 

3900 

•S9II 

5.3 

7000 

.8451 

42.5 

5100 

.7076 

►OILING-POINT. 

, K.- 

— , 

Boils  at. 

N. 

Log. 

35° 

2160 

3345 

80 

2610 

4166 

61 

3590 

5551 

46 

2350 

3711 

118 

2530 

4031 

78 

1170 

0682 

75 

2680 

4281 

56 

1720 

2355 

100 

510 

7076 

132 

6450 

8096 

182 

3220 

5079 

132 

3040 

4829 

WEIGHTS  AND  MEASURES 


23 


Table  X. 


WEIGHTS  AND  MEASURES. 

Length. 

Multiply  by  Factor. 

Given. 

Sought. 

N. 

Log. 

Inches 

Centimeters 

2.540 

40483 

Centimeters 

Inches 

.39370 

59517 

Feet 

Meters 

.30480 

48  402 

Meters 

Feet 

3.2808 

51  598 

Miles 

Kilometers 

1.6093 

20665 

Kilometers 

Miles 

Area. 

.62137 

79  335 

Sq.  Inches 

Sq.  Centimeters 

6.4516 

80  967 

Sq.  Centimeters 

Sq.  Inches 

.  1550 

19033 

Sq.  Feet 

Sq.  Meters 

.09290 

96803 

Sq.  Meters 

Sq.  Feet 

Volume. 

10.764 

03197 

Cu.  Inches 

Cu.  Centimeters 

16.387 

21450 

Cu.  Centimeters 

Cu.  Inches 

.06102 

78550 

Cu.  Feet 

Cu.  Meters 

.02832 

•45  205 

Cu.  Meters 

Cu.  Feet 

Capacity, 

35.314 

54  795 

U.  S.  Fluid  Ounces 

Cu.  Centimeters 

29.574 

.47091 

Cu.  Centimeters 

U.  S.  Fluid  Ounces 

.03381 

.52909 

U.  S.  Quarts 

Liters 

.94635 

.97605 

Liters 

U.  S.  Quarts 

1.0567 

0239s 

U.  S.  Gallons 

Liters 

3.7854 

57812 

Liters 

U.  S.  Gallons 
Weight. 

.26417 

.42  188 

Grains 

Grams 

.06480 

.81  157 

Grams 

Grains 

15.432 

.18843 

Ounces,  troy- 

Grams 

31 . 103 

.49281 

Grams 

Ounces,  troy 

.03215 

•50719 

Ounces,  av. 

Grams 

28.350 

•4525s 

Grams 

Ounces,  av. 

.03527 

•54  745 

Pounds,  av. 

Grams 

453.59 

65667 

Kilograms 

Pounds,  av. 

2.2046 

.34333 

24  NOTES  ON  THE  USE  OF  LOGARITHMS 


NOTES   ON  THE  USE  OF  LOGARITHMS. 

In  these  tables  the  ''characteristics"  of  the  logarithms,  or  the  figures  before 
their  decimal  points  which  fix  the  decimal  points  of  the  numbers,  have  been  omitted, 
because  in  ordinary  chemical  calculations  it  is  usually  easier  to  find  the  positions 
of  the  decimal  points  by  simple  inspection  of  the  numbers  employed  than  by  the 
use  of  characteristics. 

For  example,  suppose  we  have; 

.4693  g.  AgCl  (log  .67145) 
to  be  multipHed  by  the  factor  which  gives  the  amount  of  chlorine  in  it, 

.24738  (log  .39336).  '^  '^'-  ^  |i 


■7^L^.^• 


The  addition  of  the  logs  gives  the  log  .06481,  corresponding  to'  the  numerical 
figures  11610. 

Now  by  inspection,  since  .24738  is  nearly  J,  or  since  .4693  is  nearly  i,  it  is  evi- 
dent that  the  result  must  be  not  far  from  J  of  .4693  or  J  of  .24738,  so  that  the  deci- 
mal point  must  be  placed  to  correspond;  that  is,  .11610. 

Now  if  we  divide  the  amount  of  chlorine  by  the  weight  of  substance  taken  for 
analysis,  say  .2603  g.  (log  .41547)  we  have  .06481  —  .41547  =  .64934,  corresponding 
to  the  number  .44601,  the  decimal  point  of  which  is  found  by  inspection,  since  the 
result  must  be  about  4  times  .11610.  Multiplying  this  by  100  we  find  44.60%  of 
chlorine.  The  fifth  figure  in  this  result  is  an  uncertain  one  because  it  is  derived 
from  the  datum  .2603 ;  hence  it  is  best  not  to  report  it.  In  this  case  the  trouble  of 
interpolating  for  a  fifth  figure  might  have  been  avoided  by  taking  four  figures 
corresponding  to  the  nearest  logarithm. 

By  the  use  of  a  co-logarithm  (i.e.,  1  —  log)  of  the  division  the  preceding  calcu- 
lation may  be  made  by  a  single  addition : 

'^:  AgCl,  .4693  log   .67145 

^'^  Fac'or,  .24738  log   .39336 

Substance,    .2603       1  -  log   .58453 

.64934  ->  44.60% 

An  inspection  of  the  numbers  employed  above  shows  the  position  of  the  decimal 
point.     The  result  could  not  possibly  be  4.480%. 

The  student  who  is  not  skilful  in  the  use  of  logarithms  should  practice  multipli- 
cations and  divisions  with  numbers  giving  known  results  until  he  acquires  pro- 
ficiency. 

The  easily  remembered  logarithm  of  2,  which  is  .30103,  is  often  of  considerable 
assistance  in  calculations.  For  instance,  the  logarithm  of  the  atomic  weight  of 
Mg,  24.32,  is  .38596;  hence  the  log  of  2  Mg  is  .68699  and  that  of  i  Mg  is  .08493. 


LOGARITHMS  OF  NUMBERS 
FIVE-PLACE  TABLE 

WITH 

DOUBLE  THUMB-INDEX 


26 


LOGARITHMS   OF  NUMBERS. 


N. 

O 

1 

2 

3 

4 

5 

6    7 

8 

9 

P.  P. 

100 

oo  ooo 

043 

087 

130 

173 

217 

260  303 

346 

389 

101 

432 

475 

518 

561 

604 

647 

689  732 

775 

817 

1 

^«  *«j  ^a 

4  4  4 

102 

860 

903 

945 

988 

*030 

*072 

*ii5  *i57 

*i99 

*242 

2 

998 

103 

01  284 

326 

368 

410 

452 

494 

536  578 

620 

662- 

3 

13  13  13 

104 

703 

745 

787 

828 

870 

912 

953  995 

*036 

*o78 

4 
5 
6 

18  17  17 

105 

02  119 

160 

202 

243 

284 

325 

366  407 

449 

490 

26  26  25 

106 

531 

572 

612 

653 

694 

735 

776  816 

857 

898 

7 

31  30  29 

107 

938 

979 

*oi9 

*o6o 

*IOO 

*i4i 

*l8l  *222 

*262 

*302 

8 

35  34  34 

108 

03  342 

383 

423 

463 

503 

543 

583   623 

663 

703 

9 

40  39  38 

109 

743 

782 

822 

862 

902 

941 

981  *02I 

*o6o 

*IOO 

1 

110 

04  139 

179 

218 

258 

297 

336 

376   415 

454 

493 

111 

532 

571 

610 

650 

689 

727 

766   805 

844 

883 

1 

41  40  39 

4  4  4 
8  8  8 

112 

922 

961 

999 

*o38 

*o77 

*ii5 

*i54  *i92 

*231 

*269 

?, 

113 

05  308 

346 

385 

423 

461 

500 

538  576 

614 

652 

3 

12  12  12 

114 

690 

729 

767 

805 

843 

881 

918  956 

994 

*032 

4 
5 

16  16  16 
21  20  20 

115 

06  070 

108 

145 

183 

221 

258 

296.  333 

371 

408 

6 

25  24  23 

116 

446 

483 

521 

558 

595 

633 

670  707 

744 

781 

7 

29  28  27 

117 

819 

856 

893 

930 

967 

*oo4 

*o4i  *o78 

*ii5 

*i5i 

8 

33  32  31 

118 

07  188 

225 

262 

298 

335 

372 

408  445 

482 

518 

9 

37  36  35 

119 

555 

591 

628 

664 

700 

737 

773  809 

846 

882 

120 

918 

954 

990 

*027 

*o63 

*OQ9 

*i35  *i7i 

*207 

*243 

121 

08  279 

314 

350 

386 

422 

458 

493   529 

565 

600 

1 

3»  37  3» 

4  4  4 

8  7  7. 

122 

636 

672 

707 

743 

778 

814 

849  884 

920 

955 

2 

123 

991 

*026 

*o6i 

^096 

*I32 

*i67 

*202  *237 

♦272 

*307 

3 

II  II  11 

124 

09  342 

377 

412 

447 

482 

517 

552  587 

621 

656 

4 
5 
6 

15.15  14 

125 

691 

726 

760 

795 

830 

864 

899  934 

968 

*oo3 

19  19  18 
23  22  22 

126 

10  037 

072 

106 

140 

175 

209 

243  278 

312 

346 

7 

27  26  25 

127 

380 

415 

449 

483 

517 

551 

585  619 

653 

687 

8 

30  30  29 

128 

721 

755 

789 

823 

857 

890 

924  958 

992 

*025 

9 

34  33  32 

129 

II  059 

093 

126 

160 

193 

227 

261   294 

327 

361 

130 

394 

428 

461 

494 

528 

561 

594  628 

661 

694 

131 

727 

760 

793 

826 

860 

893 

926  959 

992 

*024 

1 

«d  iS^   HiS 

4  3  3 

132 

12  057 

090 

123 

i5<> 

189 

222 

254  287 

320 

352 

2 

111 

133 

385 

418 

450 

483 

516 

548 

581   613 

646 

678 

3 

II  10  10 

134 

710 

743 

775 

808 

840 

872 

905  937 

969 

*OOI 

4 
5 
6 

14  14  13 
18  17  17 

21  20  20 

135 

13  033 

066 

098 

130 

162 

194 

226  258 

290 

322 

136 

354 

386 

418 

450 

481 

513 

545   577 

609 

640 

7 

25  24  23 

137 

672 

704 

735 

767 

799 

830 

862  893 

925 

956 

8 

28  27  26 

138 

988 

*oi9 

*o5i 

*o82 

*ii4 

*i45 

*i76  *2o8 

*239 

*270 

9 

32  31  30 

139 
140 

14  301 

333 

364 

395 

426 

457 
768 

489  520 
799  829 

551 
860 

582 

891 

613 

644 

675 

706 

737 

141 

922 

953 

983 

*oi4 

*o45 

*o76 

*io6  *i37 

*i68 

*i98 

1 

<)»  <)L  OU 

3  3  3 

142 

15  229 

259 

290 

320 

351 

381 

412  442 

473 

503 

2 

6  6  6 

143 

534 

564 

594 

625 

655  . 

68s 

715   746 

776 

806 

3 

10  9  9 

144 

836 

866 

897 

927 

957 

987 

*oi7  *047 

*o77 

*io7 

4 
5 
6 

13  12  12 

145 

16  137 

167 

197 

227 

256 

286 

316  346 

376 

406 

19  19  18 

146 

435 

465 

495 

524 

554 

584 

613  643 

673 

702 

7 

22  22  21 

147 

732 

761 

791 

820 

850 

879 

909  938 

967 

997 

8 

26  25  24 

148 

17  026 

056 

085 

114 

143 

173 

202   231 

260 

289 

9 

29  28  27 

149 

319 

348 

377 

406 

435 

464 

493   522 

551 

580 

N. 

0 

1 

2 

3 

4 

5 

6    7 

8 

9 

P.  P. 

<es) 


LOGARITHMS   OF   NUMBERS. 


27 


N. 

O 

1 

<jj 

3 

4 

5 

6 

7 

8   9 

P. 

P. 

150 

17  609 

638 

667 

696 

725 

754 

782 

811 

840  869 

151 

898 

926 

955 

984 

*oi3 

*o4i 

*o7o 

*o99 

*i27  ^156 

1 

3   3 

152 

18  184 

213 

241 

270 

298 

327 

355 

384 

412  441 

2 

6   6 

153 

469 

498 

526 

554 

583 

611 

639 

667 

696  724 

3 

9   8 

154 

752 

780 

808 

837 

865 

893 

921 

949 

977  *oo5 

4 
5 
6 

12  II 

15  14 
17  17 

155 

19  033 

061 

089 

117 

145 

173 

201 

229 

257   285 

156 

312 

340 

368 

396 

424 

451 

479 

507 

535  562 

V 

20  20 

157 

590 

618 

645 

673 

700 

728 

756 

783 

811  838 

8 

2 

?  22 

158 

866 

893 

921 

948 

976 

*oo3 

*030 

*o58 

*o85  *ii2 

9 

2b   25 

159 

20  140 

167 

194 

222 

249 

276 

303 

330 

358  385 

0 

m       n£> 

160 

412 

439 

466 

493 

520 

548 

575 

602 

629  656 

161 

683 

710 

737 

763 

790 

817 

844 

871 

898  925 

1 

^    ^ 

162 

952 

978 

*oo5 

*032 

*o59 

*o85 

*II2 

*i39 

*i65  *i92 

2 

5   5 

163 

21  219 

245 

272 

299 

325 

352 

378 

405 

431  458 

3 

8   8 

164 

484 

511 

537 

564 

590 

617 

643 

669 

696  722 

4 
5 
6 

II   10 
16   16 

165 

748 

775 

801 

827 

854 

880 

906 

932 

958  985 

166 

22  on 

037 

063 

089 

115 

141 

167 

194 

220  246 

7 

I 

9  18 

167 

272 

298 

324 

350 

376 

401 

427 

453 

479   505 

8 

22   21 

168 

531 

557 

583 

608 

634 

660 

686 

712 

737   763 

9 

24   23 

169 

789 
23  045 

814 
070 

840 
096 

866 
121 

£91 
147 

917 

943 

968 

994  *oi9 

OR 

170 

172 

198 

223 

249  274 

171 

300 

325 

350 

376 

401 

426 

452 

477 

502   528 

1 

3 

172 

553 

578 

603 

629 

654 

679 

704 

729 

754  779 

2 

5 

173 

805 

830 

855 

880 

905 

930 

955 

980 

*oo5  ^030 

3 

8 

174 

24  055 

080 

105 

130 

155 

180 

204 

229 

254  279 

4 
5 
6 

10 

13 
15 

175 

304 

329 

353 

378 

403 

428 

452 

477 

502   527 

176 

551 

576 

601 

625 

650 

674 

699 

724 

748  773 

7 

18 

177 

797 

822 

846 

871 

895 

920 

944 

969 

993  *oi8 

8 

20 

178 

25  042 

066 

091 

115 

139 

164 

188 

212 

237   261 

9 

23 

179 

285 

310 

334 

358 

382 

406 

431 

455 

479   503 

0 

A           00 

180 

527 

551 

575 

600 

624 

648 

672 

696 

720   744 

181 

768 

792 

816 

840 

864 

888 

912 

935 

959  983 

1 

2    2 

182 

26  007 

031 

055 

079 

102 

126 

150 

174 

198  221 

2 

5   5 

183 

245 

269 

293 

316 

340 

364 

387 

411 

435  458 

3 

7   7- 

184 

482 

505 

529 

553 

576 

600 

623 

647 

670  694 

4 
5 
6 

K 

I 

3   9 
2  12 

\     14 

185 

717 

741 

764 

788 

811 

834 

858 

881 

905   928 

186 

951 

975 

998 

*02I 

*045 

*o68 

*09i 

*ii4 

^138  *i6i 

7 

I 

1     16 

187 

27  184 

207 

231 

254 

277 

300 

3^3 

346 

310     393 

8 

K 

?  18 

188 

416 

439 

462 

485 

508 

531 

554 

577 

600  623 

9 

2. 

2  21 

189 

646 

669 

692 

715 

738 

761 

784 

807 

830  852 

0 

3  31 

190 

875 

898 

921 

944 

967 

989 

*OI2 

*o35 

*o58  ='=081 

191 

28  103 

126 

149 

171 

194 

217 

240 

262 

285  307 

1 

' 

2   2 

1  192 

330 

353 

375 

398 

421 

443 

466 

488 

511   533 

2 

4   4  1 

193 

556 

578 

601 

623 

646 

668 

691 

713 

735   758 

3 

1       6 

194 

780 

803 

825 

847 

870 

892 

914 

937 

959  981 

4 
5 
6 

9   8 

H    II 

13  13 

195 

29  003 

026 

048 

070 

092 

115 

137 

159 

181   203 

196 

226 

248 

270 

292 

314 

336 

358 

380 

403  425 

7 

11 

)  15 

197 

447 

469 

491 

513 

535 

557 

579 

601 

623  645 

8 

it 

\     17 

198 

667 

688 

710 

732 

754 

776 

798 

820 

842  863 

9 

2C 

)  19 

199 

885 

907 

929 

951 

973 

994 
5 

*oi6 
6 

*o38 

7 

*o6o  *o8i 
8    9 

1 

0 

1 

2 

3 

4 

P. 

P. 

28 


LOGARITHMS  OF  NUMBERS. 


N. 

O 

1 

2 

3 

4t 

5 

0 

7 

8 

9 

P.P. 

200 

30  103 

125 

146 

168 

190 

211 

233 

255 

276 

298 

201 

320 

341 

363 

384 

406 

428 

449 

471 

492 

514 

1 

2       2 

202 

535 

557 

578 

600 

621 

643 

664 

68s 

707 

728 

2 

4   4 

203 

750 

771 

792 

814 

835 

856 

878 

899 

920 

942 

3 

7   6 

204 

963 

984 

*oo6 

*027 

*o48 

*o69 

^^091 

*II2 

*i33 

*i54 

4 
5 
6 

9   8 
II   II 
13  13 

205 

31  175 

197 

218 

239 

260 

281 

302 

323 

345 

366 

206 

387 

408 

429 

450 

471 

492 

513 

534 

555 

576 

7 

15  15 

207 

597 

618 

639 

660 

681 

702 

723 

744 

765 

785 

8 

18  17 

208 

806 

827 

848 

869 

890 

911 

931 

952 

973 

994 

y 

20  19 

209 

32  015 

035 

056 

077 

098 

118 

139 

160 

181 

201 

210 

222 

243 

263 

284 

305 

325 

346 

366 

387 

408 

211 

428 

449 

469 

490 

510 

531 

552 

572 

593 

613 

1 

2 

212 

634 

654 

675 

695 

715 

736 

756 

777 

797 

818 

2 

4 

213 

8sS 

858 

879 

899 

919 

940 

960 

980 

*OOI 

*02I 

3 

6 

214 

33   041 

062 

082 

102 

122 

143 

163 

183 

203 

224 

4 
5 

6 

8 
10 
12 

215 

244 

264 

284 

304 

325 

345 

365 

385 

405 

425 

216 

445 

465 

486 

506 

526 

546 

S66 

S86 

606 

626 

7 

14 

217 

646 

666 

686 

706 

726 

746 

766 

786 

806 

826 

8 

16 

218 

846 

866 

885 

905 

925 

945 

965 

985 

*oo5 

*025 

9 

18 

219 

34  044 

064 

084 

104 

124 

143 

163 

183 

203 

223 

-•  rk 

220 

242 

262 

282 

301 

321 

341 

361 

380 

400 

420 

221 

439 

459 

479 

498 

518 

537 

557 

577 

596 

616 

1 

2 

222 

635 

655 

674 

694 

713 

733 

753 

772 

792 

811 

2 

4 

223 

830 

850 

869 

889 

908 

928 

947 

967 

986 

*oo5 

3 

6 

224 

35  025 

044- 

064 

083 

102 

122 

141 

160 

180 

199 

4 
5 

6 

8 

10 
II 

225 

218 

238 

257 

276 

295 

315 

334 

353 

372 

392 

226 

411 

430 

449 

468 

488 

507 

526 

545 

564 

583 

7 

13. 

227 

603 

622 

641 

660 

679 

698 

717 

736 

755 

774 

8 

15 

228 

793 

813 

832 

851 

870 

889 

908 

927 

946 

965 

9 

17 

229 

984 

*oo3 

*02I 

*o4o 

*o59 

*o78 

*o97 

*ii6 

*i35 

*i54 

1  B 

230 

36  173 

192 

211 

229 

248 

267 

286 

305 

324 

342 

231 

361 

380 

399 

418 

436 

455 

474 

493 

511 

530 

1 

2 

232 

549 

568 

586 

605 

624 

642 

661 

680 

698 

717 

2 

4 

233 

736 

754 

773 

791 

810 

829 

847 

866. 

884 

903 

3 

5 

234 

922 

940 

959 

977 

996 

*oi4 

*o33 

*o5i 

*o7o 

*o88 

4 
5 
6 

7 

9 

II 

235 

37  107 

125 

144 

162 

181 

199 

218 

236 

254 

273 

236 

291 

310 

328 

346 

365 

383 

401 

420 

438 

457 

7 

13 

237 

475 

493 

511 

530 

548 

566 

585 

603 

621 

639 

8 
9 

14 
,6 

238 

658 

676 

694 

712 

731 

749 

767 

785 

803 

822 

239 

840 

858 

876 

894 

912 

931 

949 

967 

985 

*oo3 

■fy 

240 

^8   021 

039 

057 

075 

093 

112 

130 

148 

166 

184 

241 

202 

220 

238 

256 

274 

292 

310 

328 

346 

364 

1 

2 

242 

382 

399 

417 

435 

453 

471 

489 

507 

525 

543 

2 

3 

243 

561 

578 

596 

614 

632 

650 

668 

686 

703 

721 

3 

5 

244 

739 

757 

775 

792 

810 

828 

846 

863 

881 

899 

4 
5 
6 

7 

9 

10 

245 

917 

934 

952 

970 

987 

*oo5 

*023 

*o4i 

*o58 

*o76 

246 

39  094 

III 

129 

146 

164 

182 

199 

217 

235 

252 

7 

12 

247 

270 

287 

305 

322 

340 

358 

375 

393 

410 

428 

8 
9 

14 

248 

445 

463 

480 

498 

515 

533 

550 

568 

585 

602 

15 

249 

620 

637 

655 

672 

690 

707 

724 

742 

759 

777 

-• 

0 

1 

2 

3 

4 

5 

0 

7 

8 

9 

P.P. 

LOGARITHMS  OF  NUMBERS. 


29 


N. 
250 

0 

1 

2 

3 

4 

5    0 

7    8    9 

P.P. 

39  794 

811 

829 

846 

863 

881   898 

915  933  950 

1  a 

251 

967 

985 

*002 

*oi9 

*o37 

*o54  *o7i 

*o88  *io6  *i23 

1 

2 

252 

40  140 

157 

175 

192 

209 

226  243 

261   278   295 

2 

4 

253 

312 

32Q 

346 

364 

381 

398  415 

432  449  466 

3 

5 

254 

483 

500 

518 

535 

552 

569  586 

603   620  637 

4 
5 
6 

7 

9 

II 

255 

654 

671 

688 

705 

722 

739  756 

773   790  807 

256 

824 

841 

858 

875 

892 

909  926 

943  960  976 

7 

13 

257 

993 

*OIO 

*027 

*o44 

*o6i 

*o78  *o95 

*iii  *i28  *i45 

8 
9 

14 
16 

258 

41  162 

179 

196 

212 

229 

246  263 

280   296  313 

259 

330 

347 

363 

380 

397 

414  430 

447   464  481 

1. 

260 

497 

514 

531 

547 

564 

581  597 

614  631   647 

261 

664 

681 

697 

714 

731 

747  764 

780   797   814 

1 

2 

262 

830 

847 

863 

880 

896 

913  929 

946  963   979 

2 

3 

263 

996 

*OI2 

*029 

*o45 

*o62 

'^078  *o95 

*iii  *I27  *I44 

3 

5 

264 

42  160 

177 

193 

210 

226 

243  259 

275   292   308 

4 
5 
6 

7 

9 

10 

265 

325 

341 

357 

374 

390 

406  423 

439  455  472 

266 

488 

504 

521 

537 

553 

570  586 

602   619  635 

7 
8 

9 

12 

267 

651 

667 

684 

700 

716 

732   749 

765   781   797 

14 

268 

813 

830 

846 

862 

878 

894  911 

927  943  959 

15 

269 

975 

991 

*oo8 

*024 

*04o 

*o56  *o72 

*o88  *io4  *i2o 

lA 

270 

43  136 

152 

169 

185 

201 

217   233 

249   265   281 

271 

297 

3^3 

329 

345 

361 

377  393 

409  425  441 

1 

2 

272 

457 

473 

489 

505 

521 

537  553 

569   584  600 

2 

3 

273 

616 

632 

648 

664 

680 

696  712 

727   743   759 

3 

5 

274 
275 

775 

791 

807 

823 

838 

854  870 

*OI2  *028 

886  902   917 
*o44  *o59  *o75 

4 
5 
6 

6 

8 

10 

933 

949 

965 

981 

996 

276 

44  091 

107 

122 

138 

154 

170    185 

201   217   232 

7 
8 
9 

^^ 

277 

248 

264 

279 

295 

3" 

326   342 

358  373  389 

13 

278 

404 

420 

436 

451 

467 

483   498 

514  529  545 

14 

279 

560 

576 

592 

607 

623 

638   654 

669  685   700 

1R 

280 

716 

731 

747 

762 

778 

793  809 

824  840  855 

281 

871 

886 

902 

917 

932 

948  963 

979   994  *OIO 

1 

2 

282 

45  025 

040 

056 

071 

086 

102   117 

133   148   163 

2 

3 

283 

179 

194 

209 

225 

240 

255  271 

286  301   317 

3 

5 

284 

332 
484 

347 
500 

362 

515 

378 

530 

393 

545 

408  423 

439  454  469 

4 
5 
6 

6 

8 
9 

285 

561   576 

591   606  621 

286 

637 

652 

667 

682 

697 

712   728 

743   758  773 

7 
8 
9 

II 

287 

788 

803 

818 

834 

849 

864  879 

894  909  924 

288 

939 

954 

969 

984 

*ooo 

*or5  *o3o 

*o45  *o6o  *o75 

'4 

289 
290 

46  090 

105 

120 

135 

150 

165   180 

195  210  225 

14 

240 

255 

270 

285 

300 

3^5     330 

345  359  374 

291 

389 

404 

419 

434 

449 

464  479 

494   509   523 

1 

I 

292 

538 

553 

568 

583 

598 

613  627 

642   657   672 

2 

3 

293 

687 

702 

716 

731 

746 

761   776 

790  805   820 

3 

4 

294 

835 

850 

864 

879 

894 

909  923 

938  953  967 

4 
5 
6 

6 

7 
8 

295 

982 

997 

^^012 

*026 

*04i 

^056  *o7o 

*o85  *ioo  *ii4 

296 

47  129 

144 

159 

173 

188 

202   217 

2;^2      246   261 

7 

.  8 

q 

10 

297 

276 

290 

305 

319 

334 

349  363 

378   392   407 

13 

298 

422 

436 

451 

465 

480 

494   509 

524  538   553 

299 

567 

582 

596 

611 

625 

640   654 

669  683   698 

N. 

0 

1 

2 

3 

4 

5    6 

7    8    9 

P.P. 

30 


LOGARITHMS  OF  NUMBERS. 


N. 
300 

O 

1 

ijt 

S 

4 

5 

G 

7 

8 

0 

P.P. 

47  712 

727 

741 

756 

770 

784 

799 

813 

828 

842 

301 

857 

871 

885 

900 

914 

929 

943 

958 

972 

986 

1 

2 

302 

48  001 

015 

029 

044 

058 

073 

087 

lOI 

116 

130 

2 

3 

303 

144 

159 

173 

187 

202 

216 

230 

244 

259 

273 

3 

5 

304 

287 

302 

316 

330 

344 

359 
501 

373 

515 

387 
530 

401 

544 

416 

558 

4 
5 
6 

6 

8 
9  ' 

305 

430 

444 

458 

473 

487 

306 

572 

S86 

601 

615 

629 

643 

657 

671 

686 

700 

7 

II 

307 

714 

728 

742 

756 

770 

785 

799 

813 

827 

841 

8 

12 

308 

855 

869 

883 

897 

911 

926 

940 

954 

968 

982 

y 

H 

309 

996 

'i'OIO 

*024 

*o38 

*052 

*o66 

*o8o 

=^094 

*io8 

*I22 

310 

49  136 

150 

164 

178 

192 

206 

220 

234 

248 

262 

311 

276 

290 

304 

318 

332 

346 

360 

374 

388 

402 

312 

415 

429 

443 

457 

471 

485 

499 

513 

527 

541 

313 

554 

568 

582 

596 

610 

624 

638 

651 

665 

679 

314 

693 

707 

721 

734 

748 

762 

776 

790 

803 

817 

14 

315 

831 

845 

859 

872 

886 

900 

914 

927 

941 

955 

1 
2 

3 

316 

969 

982 

996 

*OIO 

*024 

*o37 

^051 

^065 

*o79 

*092 

3 

4 

317 

50  106 

120 

^^?> 

147 

161 

174 

188 

202 

215 

229 

4 

6 

318 

243 

256 

270 

284 

297 

311 

325 

33S 

352 

365 

5 
6 

7 
8 
9 

7 

319 

379 

393 

406 

420 

433 

447 

461 

474 

488 

501 

0 
10 
II 

320 

515 

529 

542 

556 

569 

583 

596 

610 

623 

637 

321 

651 

664 

678 

691 

705 

718 

732 

745 

759 

772 

13 

322 

786 

799 

813 

826 

840 

853 

866 

880 

893 

907 

323 

920 

934 

947 

961 

974 

987 

*OOI 

*oi4 

*028 

*04i 

324 

51  055 

068 

081 

095 

108 

121 

135 

148 

162 

175 

325 

188 

202 

215 

228 

242 

255 

268 

282 

295 

308 

326 

322 

335 

348 

362 

375 

388 

402 

415 

428 

441 

327 

455 

468 

481 

495 

508 

521 

534 

548 

561 

574 

328 

587 

601 

614 

627 

640 

654 

667 

680 

693 

706 

1 

I 

329 

720 

733 

746 

759 

772 

786 

799 

812 

825 

838 

2 
3 

4 

3 
4 

5 

330 

851 

865 

878 

891 

904 

917 

930 

943 

957 

970 

331 

983 

996 

^009 

*022 

*o35 

*o48 

*o6i 

*o75 

*o88 

*IOI 

5 

7 

332 

52  114 

127 

140 

153 

166 

179 

192 

205 

218 

231 

6 

8 

333 

244 

257 

270 

284 

297 

310 

3^3 

336 

349 

362 

7 

9 

334 

375 

388 

401 

414 

427 

440 

453 

466 

479 

492 

8 
9 

10 
12 

335 

504 

517 

530 

543 

556 

569 

582 

595 

608 

621 

336 

634 

647 

660 

673 

686 

699 

711 

724 

737 

750 

337 

763 

776 

789 

802 

815 

827 

840 

853 

866 

879 

338 

892 

905 

917 

930 

943 

956 

969 

982 

994 

*oo7 

339 
340 

53  020 

033 

046 

058 

071 

084 

097 

no 

122 

135 

148 

161 

173 

186 

199 

212 

224 

237 

250 

263 

341 

275 

288 

301 

314 

326 

339 

352 

364 

377 

390 

1 

I 

342 

403 

415 

428 

441 

453 

466 

479 

491 

504 

517 

2 

2 

343 

529 

542 

555 

567 

580 

593 

605 

618 

631 

643 

3 

4 

344 

656 

668 

681 

694 

706 

719 

732 

744 

757 

769 

4 
5 
6 

5 
6 

7 

345 

782 

794 

807 

820 

832 

845 

857 

870 

882 

895 

346 

908 

920 

933 

945 

958 

970 

983 

995 

*oo8 

*020 

7 

8 

347 

54  033 

045 

058 

070 

083 

095 

108 

120 

133 

145 

8 
9 

10 

348 

158 

170 

183 

195 

208 

220 

233 

245 

258 

270 

349 

283 

295 

307 

320 

332 

345 

357 

370 

382 

394 

N.  ■ 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

P.P. 

1 

LOGARITHMS   OF   NUMBERS. 


31 


N. 

O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

P.P. 

350 

54  407 

419 

432 

444 

456 

469 

481 

494 

506 

518 

351 

531 

543 

555 

568 

580 

593 

605 

617 

630 

642 

1 

1 

352 

654 

667 

679 

691 

704 

716 

728 

741 

753 

765 

2 

3 

353 

777 

790 

802 

814 

827 

839 

851 

864 

876 

888 

3 

4 

354 

900 

913 

925 

937 

949 

962 

974 

986 

998 

*OII 

4 
5 
6 

5 

7 
8 

355 

55  023 

035 

047 

060 

072 

084 

096 

108 

121 

^33 

356 

145 

157 

169 

182 

194 

206 

218 

230 

242 

255 

7 

9 

357 

267 

279 

291 

303 

315 

328 

340 

352 

364 

376 

8 
9 

10 

358 

388 

400 

413 

425 

437 

449 

461 

473 

485 

497 

359 

509 

522 

534 

546 

558 

570 

582 

594 

606 

618 

360' 

630 

642 

654 

666 

678 

691 

703 

715 

727 

739 

361 

751 

763 

775 

787 

799 

811 

823 

835 

847 

859 

362 

871 

883 

895 

907 

919 

931 

943 

955 

967 

979 

363 

991 

*oo3 

*oi5 

*027 

*o38 

*o5o 

*o62 

*o74 

*o86 

*098 

364 

56  no 

122 

134 

146 

158 

170 

182 

194 

205 

217 

1  ^^ 

365 

229 

241 

253 

265 

277 

289 

301 

312 

324 

336 

1 
2 

I 
2 

366 

348 

360 

372 

384 

396 

407 

419 

431 

443 

455 

3 

4 

367 

467 

478 

490 

502 

514 

526 

538 

549 

561 

573 

4 

5 

368 

585 

597 

608 

620 

632 

644 

656 

667 

679 

691 

5 

6 

369 

703 

714 

726 

738 

750 

761 

773 

785 

797 

808 

6 

7 
8 

7 

8 

10 

370 

820 

832 

844 

855 

867 

879 

891 

902 

914 

926 

371 

937 

949 

961 

972 

984 

996 

*oo8 

*oi9 

*o3i 

*o43 

9 

II 

372 

57  054 

066 

078 

089 

lOI 

113 

124 

136 

148 

159 

373 

171 

183 

194 

206 

217 

229 

241 

252 

264 

276 

374 

287 

299 

310 

322 

334 

345 

357 

368 

380 

392 

375 

403 

415 

426 

438 

449 

461 

473 

484 

496 

507 

376 

519 

530 

542 

553 

565 

576 

588 

600 

611 

623 

377 

634 

646 

657 

669 

680 

692 

703 

715 

726 

738 

-•  -1 

378 

749 

761 

772 

784 

795 

807 

818 

830 

841 

852 

1 

I 

379 

864 

875 

887 

898 

910 

921 

933 

944 

955 

967 

2 
3 

4 

2 

3 

4 

380 

978 

990 

*OOI 

*oi3 

*024 

*o35 

*047 

*o58 

*o7o 

*o8i 

381 

58  092 

104 

115 

127 

138 

149 

161 

172 

184 

195 

5 

6 

382 

206 

218 

229 

240 

252 

263 

274 

286 

297 

309 

6 

7 

383 

320 

33^ 

343 

354 

365 

377 

388 

399 

410 

422 

7 

8 

384 

433 

444 

456 

467 

478 

490 

501 

512 

524 

535 

8 
9 

9 

385 

546 

557 

569 

580 

591 

602 

614 

625 

636 

647 

386 

659 

670 

681 

692 

704 

715 

726 

737 

749 

760 

387 

771 

782 

794 

805 

816 

827 

838 

850 

861 

872 

388 

883 

894 

906 

917 

928 

939 

950 

961 

973 

984 

389 

995 

*oo6 

*oi7 

*028 

^^040 

*o5i 

*o62 

*o73 

*o84 

*095 

390 

19  106 

•118 

129 

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151 

162 

173 

184 

195 

207 

391- 

:'■;   2i8 

229 

240 

251 

262 

273 

284 

295 

306 

318' 

1 

I 

393. 

•     329 

■.340 

351 

362 

373 

384 

395 

406 

417 

428 

2 

2 

393 

439 

r.450 

461 

472 

483 

494 

506 

517 

5'28 

539 

3 

3 

394 
395 

550 

.561 

572 

583 

594 

605 
715 

616 
726 

627 

737 

638 

748 

649 
759 

4 
5 
6 

4 
5 
6 

660 

671 

682 

693 

704 

396 

770 

780 

791 

802 

813 

824 

835 

846 

857 

868 

/ 

7 

397 

879 

890 

901 

912 

923 

934 

945 

956 

966 

977 

8 

8 

398 

988 

999 

*OIO 

*02I 

*032 

■1=043 

*o54 

*o65 

^076 

*o86 

9 

9 

399 

60  097 

108 

119 

130 

141 

152 

163 

173 

184 

195 

N. 

0 

■ 

1 

2 

3 

4 

5 

6 

7 

8 

9 

P.P. 

32 


LOGARITHMS  OF  NUMBERS. 


N. 

O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

P.P. 

400 

6o  2o6 

217 

228 

239 

249 

260 

271 

282 

293 

304 

401 

314 

325 

336 

347 

358 

369 

379 

390 

401 

412 

402 

423 

433 

444 

455 

466 

477 

487 

498 

509 

520 

403 

531 

541 

552 

563 

574 

584 

595 

606 

617 

627 

404 

638 

649 

660 

670 

681 

692 

703 

713 

724 

735 

1 1 

405 

746 

756 

767 

778 

788 

799 

810 

821 

831 

842 

406 

853 

863 

874 

885 

895 

906 

917 

927 

938 

949 

I 

407 

959 

970 

981 

991 

*002 

*oi3 

*023 

*o34 

*o45 

*o55 

2 

2 

408 

61  066 

077 

087 

098 

109 

119 

130 

140 

151 

162 

3 

3 

409 

172 

183 

194 

204 

215 

225 

236 

247 

257 

268 

4 
5 
6 

4 
6 

7 

410 

278 

289 

300 

310 

321 

33^ 

342 

352 

363 

374 

411 

384 

395 

405 

416 

426 

437 

448 

458 

469 

479 

,  7 
8 
9 

8 

412 

490 

500 

511 

521 

532 

542 

553 

563 

574 

584 

9 

413 

595 

606 

616 

627 

637 

648 

658 

669 

679 

690 

414 

700 

711 

721 

731 

742 

752 

763 

773 

784 

794 

415 

805 

815 

826 

836 

847 

857 

868 

878 

888 

899 

416 

909 

920 

930 

941 

951 

962 

972 

982 

993 

*oo3 

417 

62  014 

024 

034 

045 

055 

066 

076 

086 

097 

107 

418 

118 

128 

138 

149 

159 

170 

180 

190 

201 

211 

419 

221 

232 

242 

252 

263 

273 

284 

294 

304 

315 

420 

325 

335 

346 

356 

366 

377 

387 

397 

408 

418 

421 

428 

439 

449 

459 

469 

480 

490 

500 

511 

521 

1 

lU 

422 

531 

542 

552 

562 

572 

583 

593 

603 

613 

624 

2 

2 

423 

634 

644 

655 

66s 

675 

685 

696 

706 

716 

726 

3 

3 

424 

737 

747 

757 

767 

778 

788 

798 

808 

818 

829 

4 
5 
6 

4 
5 
6 

425 

839 

849 

859 

870 

880 

890 

900 

910 

921 

931 

426 

941 

951 

961 

972 

982 

992 

*002 

*OI2 

*022 

"^^33 

7 

7 

427 

63  043 

053 

063 

073 

083 

094 

104 

114 

124 

134 

8 

8 

428 

144 

155 

165 

175 

185 

195 

205 

215 

225 

236 

9 

9 

429 

246 

256 

266 

276 

286 

296 

306 

317 

327 

337 

430 

347 

357 

367 

377 

387 

397 

407 

417 

428 

438 

431 

448 

458 

468 

478 

488 

498 

508 

518 

528 

538 

432 

548 

558 

568 

579 

589 

599 

609 

619 

629 

639 

433 

649 

659 

669 

679 

689 

699 

709 

719 

729 

739 

434 

749 

759 

769 

779 

789 

799 

809 

819 

829 

839 

n 

435 

849 

859 

869 

879 

889 

899 

909 

919 

929 

939 

436 

949 

959 

969 

979 

988 

998 

*oo8 

*oi8 

*028 

*o38 

1 

I 

437 

64  048 

058 

068 

078 

088 

098 

108 

118 

128 

^37 

2 

2 

438 

147 

157 

167 

177 

187 

197 

207 

217 

227 

237 

3 

3 

439 

246 

256 

266 

276 

286 

296 

306 

316 

326 

335 

4 
5 
6 

7- 

8 

9 

4 

5 

■  5 

6 

440 

345 

355 

365 

375 

385 

395 

404 

414 

424 

434  * 

441 

444 

454 

464 

473 

483 

493 

503 

513 

523 

532 

442 

542 

552 

562 

572 

582 

591 

601 

611 

621 

631 

.  7 
8 

443 

640 

650 

660 

670 

680 

689 

699 

709 

719 

729 

444 

738 

748 

758 

768 

777 

787 

797 

807 

816 

826 

445 

836 

846 

856 

865 

875 

885 

895 

904 

914 

924 

446 

933 

943 

953 

963 

972 

982 

992 

*002 

*OII 

*02I 

447 

65  031 

040 

050 

060 

070 

079 

089 

099 

108 

118  . 

448 

128 

137 

147 

157 

167 

176 

186 

196 

205 

215 

449 

225 

234 

244 

254 

263 

273 

283 

292 

302 

312 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

P.P. 

LOGARITHMS  OF  NUMBERS. 


33 


N. 

O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

P. 

P. 

450 

65  321 

331 

341 

350 

360 

369 

379 

389 

398 

408 

451 

418 

427 

437 

447 

456 

466 

475 

485 

495 

504 

452 

514 

523 

533 

543 

552 

562 

571 

581 

591 

600 

453 

610 

619 

629 

639 

648 

658 

667 

677 

686 

696 

454 

706 

715 

725 

734 

744 

753 

763 

772 

782 

792 

455 

801 

811 

820 

830 

839 

849 

858 

868 

877 

887 

10 

456 

896 

906 

916 

925 

935 

944 

954 

963 

973 

982 

1 

I 

457 

992 

*OOI 

*OII 

*020 

*o3o 

*039 

*o49 

*o58 

*o68 

*o77 

2 

2 

458 

66  087 

096 

106 

115 

124 

134 

143 

153 

162 

172 

3 
4 
5 
6 

7 
8 

3 

459 

181 

191 

200 

210 

219 

229 

238 

247 

257 

266 

4 
5 
6 

460 

276 

285 

295 

304 

314 

323 

332 

342 

351 

361 

461 

370 

380 

389 

398 

408 

417 

427 

436 

445 

455 

7 
8 

462 

464 

474 

483 

492 

502 

511 

521 

530 

539 

549 

9 

Q 

463 

558 

567 

577 

S86 

596 

605 

614 

624 

633 

642 

464 

652 

745 

661 

755 

671 
764 

680 

773 

689 

783 

699 

708 

717 

727 

736 

465 

792 

801 

811 

820 

829 

466 

839 

848 

857 

867 

876 

885 

894 

904 

913 

922 

467 

932 

941 

950 

960 

969 

978 

987 

997 

*oo6 

*oi5 

468 

67  025 

034 

043 

052 

062 

071 

080 

089 

099 

108 

469 

117 

127 

136 

145 

154 

164 

173 

182 

191 

201 

9 

470 

210 

219 

228 

237 

247 

256 

265 

274 

284 

293 

471 

302 

3" 

321 

330 

339 

348 

357 

367 

376 

385 

1 

I 

472 

394 

403 

413 

422 

431 

440 

449 

459 

468 

477 

2 

2 

473 

486 

495 

504 

514 

523 

532 

541 

550 

560 

569 

3 

3 

474 

578 

587 

596 

605 

614 

624 

^33 

642 

651 

660 

4 
5 
6 

7 
8 
9 

4 
5 
5 
6 

475 

669 

679 

688 

697 

706 

715 

724 

733 

742 

752 

476 

761 

770 

779 

788 

797 

806 

815 

825 

834 

843 

477 

852 

861 

870 

879 

888 

897 

906 

916 

925 

934 

7 
8 

478 

943 

952 

961 

970 

979 

988 

997 

*oo6 

*oi5 

*024 

479 
480 

68  034 

043 

052 

061 

070 

079 
169 

088 
178 

097 
187 

106 
196 

115 
205 

124 

^^3 

142 

151 

160 

481 

215 

224 

233 

242 

251 

260 

269 

278 

287 

296 

482 

305 

314 

323 

332 

341 

350 

359 

368 

377 

386 

483 

395 

404 

413 

422 

431 

440 

449 

458 

407 

476 

484 

485 

494 

502 

511 

520 

529 

538 

547 

556 

565 

g 

485 

574 

583 

592 

601 

610 

619 

628 

637 

646 

655 

486 

664 

673 

681 

690 

699 

708 

717 

726 

735 

744 

1 

487 

753 

762 

771 

780 

789 

797 

806 

815 

824 

833 

2 

2 

488 

842 

851 

860 

869 

878 

886 

895 

904 

913 

922 

3 

2 

489 

931 

940 

949 

958 

966 

975 

984 

993 

*002 

*OII 

4 
5 
6 

3 
4 
5 

490 

69  020 

028 

037 

046 

055 

064 

073 

082 

090 

099 

491 

108 

117 

126 

135 

144 

152 

161 

170 

179 

188 

7 
8 
9 

6 
6 

492 

197 

205 

214 

223 

232 

241 

249 

258 

267 

276 

493 

285 

294 

302 

311 

320 

329 

33» 

346 

355 

364 

7 

494 

373 

381 

390 

399 

408 

417 

425 

434 

443 

452 

495 

461 

469 

478 

487 

496 

504 

513 

522 

531 

539 

496 

548 

557 

566 

574 

583 

592 

601 

609 

618 

627 

497 

636 

644 

653 

662 

671 

679 

688 

697 

705 

714 

498 

723 

732 

740 

749 

758 

767 

775 

784 

793 

801 

499 

810 

819 

827 

836 

845 

854 

862 

871 

880 

8S8 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

P. 

P. 

34 


LOGARITHMS   OF  NUMBERS. 


N. 

O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

P.P. 

500 

69  897 

906 

914 

923 

932 

940 

949 

958 

966 

975 

501 

984 

992 

*OOI 

*OIO 

*oi8 

*027 

*036 

*044 

*o53 

*o62 

502 

70  070 

079 

088 

096 

105 

114 

122 

131 

140 

148 

503 

157 

165 

174 

i«3 

191 

200 

209 

217 

226 

234 

504 

243 

252 

260 

269 

278 

286 

295 

303 

312 

321 

505 

329 

338 

346 

355 

364 

372 

381 

389 

398 

406 

506 

415 

424 

432 

441 

449 

458 

467 

475 

484 

492 

1 

» 

507 

501 

509 

518 

526 

535 

544 

552 

561 

569 

578 

2 

2 

508 

586 

595 

603 

612 

621 

629 

638 

646 

655 

663 

3 

3- 

509 

672 

680 

689 

697 

706 

714 

723 

731 

740 

749 

4 
5 
6 

4 
5 
5 

510 

757 

766 

774 

783 

791 

800 

808 

817 

825 

834 

511 

842 

«5i 

859 

868 

876 

885 

893 

902 

910 

919 

7 

6 

512 

927 

935 

944 

952 

961 

969 

978 

986 

995 

*oo3 

8 

7 

513 

71  012 

020 

029 

037 

046 

054 

063 

071 

079 

088 

9 

8 

514 

096 

105 

"3 

122 

130 

139 

147 

155 

164 

172 

515 

181 

189 

198 

206 

214 

223 

231 

240 

248 

257 

516 

265 

273 

282 

290 

299 

307 

315 

324 

332 

341 

517 

349 

357 

366 

374 

383 

391 

399 

408 

416 

425 

518 

433 

441 

450 

458 

466 

475 

483 

492 

500 

508 

519 

517 

525 

533 

542 

550 

559 

567 

575 

584 

592 

520 

600 

609 

617 

625 

634 

642 

650 

659 

667 

675 

521 

684 

692 

700 

709 

717 

725 

734 

742 

750 

759 

1 

8 

522 

767 

775 

784 

792 

800 

809 

817 

825 

834 

842 

2 

2 

523 

850 

858 

867 

875 

883 

892 

900 

908 

917 

925 

3 

2 

524 

933 

941 

950 

958 

966 

975 

983 

991 

999 

*oo8 

4 
5 
6 

3 
4 
5 

525 

72  016 

024 

032 

041 

049 

057 

066 

074 

082 

090 

526 

099 

107 

115 

123 

132 

140 

148 

156 

165 

173 

7 

6 

527 

181 

189 

198 

206 

214 

222 

230 

239 

247 

255 

8 

6 

528 

263 

272 

280 

288 

296 

304 

313 

321 

329 

337 

9 

7 

529 

346 

354 

362 

370 

378 

387 

395 

403 

411 

419 

530 

428 

436 

444 

452 

460 

469 

477 

485 

493 

501 

531 

509 

518 

526 

534 

542 

550 

558 

567 

575 

583 

532 

591 

599 

607 

616 

624 

632 

640 

648 

6s6 

665 

533 

673 

681 

689 

697 

705 

713 

722 

730 

738 

746 

534 

754 

762 

770 

779 

787 

795 

803 

811 

819 

827 

ti 

535 

835 

843 

852 

860 

868 

876 

884 

892 

900 

908 

536 

916 

925 

933 

941 

949 

957 

965 

973 

981 

989 

1 

I 

537 

997 

*oo6 

*oi4 

*022 

*o3o 

*o38 

*o46 

*o54 

*o62 

*o7o 

2 

I 

538 

73  078 

086 

094 

102 

III 

119 

127 

135 

143 

151 

3 

2 

539 

159 

167 

175 

183 

191 

199 

207 

215 

223 

231 

4 
5 
6 

3 
4 
4 

540 

239 

247 

255 

263 

272 

280 

288 

296 

304 

312 

541 

320 

328 

336 

344 

352 

360 

368 

376 

384 

392 

7 

5 

542 

400 

408 

416 

424 

432 

440 

448 

456 

464 

472 

8 
9 

6 
6 

543 

480 

488 

496 

504 

512 

520 

528 

536 

544 

552 

544 

560 

568 

576 

584 

592 

600 

608 

616 

624 

632 

545 

640 

648 

656 

664 

672 

679 

687 

695 

703 

711 

546 

719 

727 

735 

743 

751 

759 

767 

775 

783 

791 

547 

799 

807 

815 

823 

830 

838 

846 

854 

862 

870 

548 

878 

886 

894 

902 

910 

918 

926 

933 

941 

949 

549 

957 

965 

973 

981 

989 

997 

*oo5 

*oi3 

*020 

*028 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

P.  P. 

LOGARITHMS  OF  NUMBERS. 


35 


K. 

O 

1 

2 

3 

4 

5 

0 

7 

8 

0 

P.P. 

550 

74  036 

044 

052 

060 

068 

076 

084 

092 

099 

107 

551 

115 

123 

131 

139 

147 

155 

162 

170 

178 

186 

552 

194 

202 

210 

218 

225 

233 

241 

249 

257 

265 

553 

273 

280 

288 

296 

304 

312 

320 

327 

335 

343 

554 

351 

359 

367 

374 

382 

390 

398 

406 

414 

421 

555 

429 

437 

445 

453 

461 

468- 

476 

484 

492 

500 

556 

507 

515 

523 

531 

539 

547 

554 

562 

570 

578 

557 

586 

593 

601 

609 

617 

624 

632 

640 

648 

656 

558 

663 

671 

679 

687 

695 

702 

710 

718 

726 

733 

559 

741 

749 

757 

764 

772 

780 
858 

788 
865 

796 
873 

803 
881 

811 
889 

ft 

560 

819 

827 

834 

842 

850 

561 

896 

904 

912 

920 

927 

935 

943 

950 

958 

966 

1 

I 

562 

974 

981 

989 

997 

*oo5 

*OI2 

*020 

*028 

*035 

*043 

2 

2 

563 

75  051 

059 

066 

074 

082 

089 

097 

105 

113 

120 

3 

2 

564 

128 

136 

143 

151 

159 

166 
243 

174 
251 

182 
259 

189 
266 

197 

274 

4 
5 
6 

3 
4 

5 

565 

205 

213 

220 

228 

236 

566 

282 

289 

297 

305 

312 

320 

328 

335 

343 

351 

7 
8 
q 

6 
6 

7 

567 

358 

366 

374 

381 

389 

397 

404 

412 

420 

427 

568 

435 

442 

450 

458 

465 

473 

481 

488 

496 

504 

569 

511 

519 

526 

534 

542 

549 

557 

565 

572 

580 

570 

587 

595 

603 

610 

618 

626 

633 

641 

648 

656 

571 

664 

671 

679 

686 

694 

702 

709 

717 

724 

732 

572 

740 

747 

755 

762 

770 

778 

785 

793 

800 

808 

573 

815 

823 

831 

838 

846 

853 

861 

868 

876 

884 

574 

891 

899 

906 

914 

921 

929 

937 

944 

952 

959 

575 

967 

974 

982 

989 

997 

*oo5 

*OI2 

*020 

*027 

*035 

576 

76  042 

050 

057 

065 

072 

080 

087 

095 

103 

no 

577 

118 

125 

^33 

140 

148 

155 

163 

170 

178 

185 

578 

193 

200 

208 

215 

223 

230 

238 

245 

253 

260 

579 

268 

275 

283 

290 

298 

305 

3^3 

320 

328 

335 

■7 

580 

343 

350 

358 

365 

373 

380 

388 

395 

403 

410 

581 

418 

425 

433 

440 

448 

455 

462 

470 

477 

485 

1 

I 

582 

492 

500 

507 

515 

522 

530 

537 

545 

552 

559 

2 

I 

583 

567 

574 

582 

589 

597 

604 

612 

619 

626 

634 

3 

2 

584 

641 

649 

656 

664 

671 

678 

686 

693 

701 

708 

4 
5 
6 

3 
4 
4 

585 

716 

723 

730 

738 

745 

753 

760 

768 

775 

782 

586 

790 

797 

805 

812 

819 

827 

834 

842 

849 

856 

7 

5 

587 

864 

871 

879 

886 

893 

901 

908 

916 

923 

930 

8 
9 

6 
6 

588 

938 

945 

953 

960 

967 

975 

982 

989 

997 

*oo4 

589 

77  012 

019 

026 

034 

041 

048 

056 

063 

070 

078 

590 

085 

093 

100 

107 

115 

122 

129 

137 

144 

151 

591 

159 

166 

173 

181 

188 

195 

203 

210 

217 

225 

592 

232 

240 

247 

254 

262 

269 

276 

283 

291 

298 

593 

305 

3-^3 

320 

327 

335 

342 

349 

357 

364 

371 

594 

379 

386 

393 

401 

408 

415 

422 

430 

437 

444 

595 

452 

459 

466 

474 

481 

488 

495 

503 

510 

517 

596 

525 

532 

539 

546 

554 

561 

568 

576 

583 

590 

597 

597 

605 

612 

619 

627 

634 

641 

648 

6s6 

663 

598 

670 

677 

685 

692 

699 

706 

714 

721 

728 

735 

599 

743 

750 

757 

764 

772 

779 
5 

786 
6 

793 
7 

801 
8 

8c8 
9 

N. 

0 

1 

2 

3 

4 

P.P. 

36 


LOGARITHMS   OF  NUMBERS. 


N. 

O 

1 

2 

3 

4 

5 

6 

7 

8 

i) 

P. 

P. 

600 

77  815 

822 

830 

837 

844 

851 

859 

866 

873 

880 

601 

887 

895 

902 

909 

916 

924 

931 

938 

945 

952 

602 

960 

967 

974 

981 

988 

996 

*oo3 

*OIO 

*oi7 

*025 

603 

78  032 

039 

046 

053 

061 

068 

075 

082 

089 

097 

604 

104 

III 

118 

125 

132 

140 

147 

154 

161 

168 

605 

176 

183 

190 

197 

204 

211 

219 

226 

233 

240 

606 

247 

254 

262 

269 

276 

283 

290 

297 

305 

312 

1 

8 
I 

607 

319 

326 

333 

340 

347 

355 

362 

369 

376 

383 

2 

2 

608 

390 

398 

405 

412 

419 

426 

433 

440 

447 

455 

3 

2 

609 

462 

469 

476 

483 

490 

497 

504 

512 

519 

526 

4 
5 
6 

3 
4 
5 

610 

533 

540 

547 

554 

561 

569 

576 

583 

590 

597 

611 

604 

611 

618 

625 

633 

640 

647 

654 

661 

668 

7 

6 

612 

675 

682 

689 

696 

704 

711 

718 

725 

732 

739 

8 

6 

613 

746 

753 

760 

767 

774 

781 

789 

796 

803 

810 

9 

7 

614 

817 

824 

831 

838 

845 

852 
923 

859 
930 

866 
937 

873 
944 

880 
951 

615 

888 

895 

902 

909 

916 

616 

958 

965 

972 

979 

986 

993 

*ooo 

*oo7 

*oi4 

*02I 

617 

79  029 

036 

043 

050 

057 

064 

071 

078 

085 

092 

618 

099 

106 

113 

120 

127 

134 

141 

148 

155 

162 

619 

169 

176 

183 

190 

197 

204 

211 

218 

225 

232 

620 

239 

246 

253 

260 

267 

274 

281 

288 

295 

302 

621 

309 

316 

323 

33° 

337 

344 

351 

358 

365 

372 

1 

7 

I 

622 

379 

386 

393 

400 

407 

414 

421 

428 

435 

442 

2 

I 

623 

449 

456 

463 

470 

477 

484 

491 

498 

505 

5" 

3 

2 

624 

518 

525 

532 

539 

546 

553 

560 

567 

574 

581 

4 
5 
6 

3 
4 
4 

625 

588 

595 

602 

609 

616 

623 

630 

637 

644 

650 

626 

657 

664 

671 

678 

685 

692 

699 

706 

713 

720 

7 

5 

627 

727 

734 

741 

748 

754 

761 

768 

775 

782 

789 

8 

6 

628 

796 

803 

810 

817 

824 

831 

837 

844 

851 

858 

9 

6 

629 

865 

872 

879 

886 

893 

900 
969 

906 

975 

913 
982 

920 
989 

927 
996 

630 

934 

941 

948 

955 

962 

631 

80  003 

010 

017 

024 

030 

037 

044 

051 

058 

065 

632 

072 

079 

085 

092 

099 

106 

113 

120 

127 

134 

633 

140 

147 

154 

161 

168 

175 

182 

188 

195 

202 

634 

209 

216 

223 

229 

236 

243 

250 

257 

264 

271 

635 

277 

284 

291 

298 

305 

312 

318 

325 

332 

339 

636 

346 

353 

359 

366 

373 

380 

387 

393 

400 

407 

1 

6 

637 

414 

421 

428 

434 

441 

448 

455 

462 

468 

475 

2 

I 

638 

482 

489 

496 

502 

509 

516 

523 

530 

536 

543 

3 

2 

639 

550 

557 

564 

570 

577 

584 

591 

598 

604 

611 

4 
5 

6 

2 

3 
4 

640 

618 

625 

632 

638 

645 

652 

659 

665 

672 

679 

641 

686 

693 

699 

706 

713 

720 

726 

733 

740 

747 

7 

4 

642 

754 

760 

767 

774 

781 

787 

794 

801 

808 

814 

8 

5 

643 

821 

828 

835 

841 

848 

855 

862 

868 

875 

882 

9 

5 

644 

889 

895 

902 

909 

916 

922 

929 

936 

943 

949 

645 

956 

963 

969 

976 

983 

990 

996 

*oo3 

*OIO 

*oi7 

646 

81  023 

030 

037 

043 

050 

057 

064 

070 

077 

084 

647 

090 

097 

104 

III 

117 

124 

131 

137 

144 

151 

648 

158 

164 

171 

178 

184 

191 

198 

204 

211 

218 

649 

224 

231 

238 

245 

251 

258 

265 

271 

278 

285 

P. 

P. 

X. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

LOGARITHMS  OF  NUMBERS. 


37 


N. 

O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

P.P. 

650 

8i  291 

298 

305 

311 

318 

325 

33^ 

338 

345 

351 

651 

35& 

365 

371 

378 

385 

391 

398 

405 

411 

418 

652 

425 

431 

438 

445 

451 

458 

465 

471 

478 

485 

653 

491 

498 

505 

511 

518 

525 

531 

53^ 

544 

551 

654 
655 

558 

564 

571 

578 

584- 

591 
657 

598 
664 

604 
671 

611 
677 

617 
684 

624 

631 

637 

644 

651 

656 

690 

697 

704 

710 

717 

723 

730 

737 

743 

750 

657 

757 

763 

770 

776 

783 

790 

796 

803 

809 

816 

658 

823 

829 

836 

842 

849 

856 

862 

869 

875 

882 

659 

889 

895 

902 

908 

915 

921 

928 

935 

941 

948 

660 

954 

961 

968 

974 

981 

987 

994 

*ooo 

*oo7 

*oi4 

661 

82  020 

027 

033 

040 

046 

053 

060 

066 

073 

079 

1 

I 

662 

086 

092 

099 

105 

112 

119 

125 

132 

138 

145 

2 

I 

663 

151 

158 

164 

171 

178 

184 

191 

197 

204 

210 

3 

2 

664 

217 

223 

230 

236 

243 

249 

256 

263 

269 

276 

4 

5 
6 

3 
4 
4 

665 

282 

289 

295 

302 

308 

315 

321 

328 

334 

341 

666 

347 

354 

360 

367 

373 

380 

387 

393 

400 

406 

7 

5 

667 

413 

419 

426 

432 

439 

445 

452 

458 

465 

471 

8 
9 

6 

668 

478 

484 

491 

497 

504 

510 

517 

523 

530 

536 

6 

669 

543 

549 

556 

562 

569 

575 

582 

588 

595 

601 

670 

607 

614 

620 

627 

633 

640 

646 

653 

659 

666 

671 

672- 

679 

685 

692 

698 

705 

711 

718 

724 

730 

672 

737 

743 

750 

756 

763 

769 

776 

782 

789 

795 

673 

802 

808 

814 

821 

827 

834 

840 

847 

853 

860 

674 

866 

872 

879 

885 

892 

898 

905 

911 

918 

924 

675 

930 

937 

943 

950 

956 

963 

969 

975 

982 

988 

676 

995 

*OOI 

*oo8 

*oi4 

*020 

*027 

*033 

*040 

*o46 

*052 

677 

83  059 

065 

072 

078 

085 

091 

097 

104 

no 

117 

678 

123 

129 

136 

142 

149 

155 

161 

168 

174 

181 

679 

187 

193 

200 

206 

213 

219 

283 

225 
289 

232 
296 

238 
302 

245 
308 

ti 

680 

251 

257 

264 

270 

276 

681 

315 

321 

327 

334 

340 

347 

353 

359 

366 

372 

1 

I 

682 

378 

385 

391 

398 

404 

410 

417 

423 

429 

436 

2 

I 

683 

442 

448 

455 

461 

467 

474 

480 

487 

493 

499 

3 

2 

684 

506 

512 

518 

525 

53^ 

537 

544 

550 

556 

563 

4 
5 
6 

2 

3 
4 

685 

569 

575 

582 

S88 

594 

601 

607 

613 

620 

626 

686 

632 

639 

645 

651 

658 

664 

670 

677 

683 

689 

7 

4 

687 

696 

702 

708 

715 

721 

727 

734 

740 

746 

753 

8 

5 

688 

759 

765 

771 

778 

784 

790 

797 

803 

809 

816 

9 

5 

689 

822 

828 

835 

841 

847 

853 

860 

866 

872 

879 

690 

885 

891 

897 

904 

910 

916 

923 

929 

935 

942 

691 

948 

954 

960 

967 

973 

979 

985 

992 

998 

*oo4 

692 

84  on 

017 

023 

029 

036 

042 

048 

055 

061 

067 

693 

073 

080 

086 

092 

098 

105 

III 

117 

123 

130 

694 

136 

142 

148 

155 

161 

167 
230 

173 
236 

180 
242 

186 
248 

192 

255 

695 

198 

205 

211 

217 

223 

696 

261 

267 

273 

280 

286 

292 

298 

305 

311 

317 

697 

323 

330 

336 

342 

348 

354 

361 

367 

373 

379 

698 

386 

392 

398 

404 

410 

417 

423 

429 

435 

442 

699 

448 

454 

460 

466 

473 

479 

485 

491 

497 

504 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

P.P. 

38 


LOGARITHMS   OF   NUMBERS. 


». 

O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

P.P. 

;700 

84  510 

516 

522 

528 

535 

541 

547 

553 

559 

566 

701 

572 

57« 

5«4 

590 

597 

603 

609 

615 

621 

628 

702 

634 

640 

646 

652 

658 

665 

671 

677 

683 

689 

703 

696 

702 

708 

714 

720 

726 

733 

739 

745 

751 

704 

757 

763 

770 

776 

782 

788 

794 

800 

807 

813 

705 

819 

825 

831 

837 

844 

850 

856 

862 

868 

874 

706 

880 

887 

893 

899 

905 

911 

917 

924 

930 

936 

1 

7 
I 

707 

942 

948 

954 

960 

967 

973 

979 

985 

991 

997 

2 

I 

708 

85  003 

009 

016 

022 

028 

034 

040 

046 

052 

058 

3 

2 

709 

065 

071 

077 

083 

089 

095 

lOI 

107 

114 

120 

4 
5 

6 

3 
4 
4 

710 

126 

132 

138 

144 

150 

156 

163 

169 

175 

181 

711 

187 

193 

199 

205 

211 

217 

224 

230 

236 

242 

7 

5 

712 

248 

254 

260 

266 

272 

278 

285 

291 

297 

3^3 

8 

6 

713 

309 

315 

321 

327 

333 

339 

345 

352 

3!^S 

364 

9 

6 

714 

370 

376 

382 

388 

394 

400 

406 

412 

418 

425 

715 

431 

437 

443 

449 

455 

461 

467 

473 

479 

485 

716 

491 

497 

503 

509 

516 

522 

528 

534 

540 

546 

717 

552 

558 

564 

570 

576 

582 

588 

594 

600 

606 

718 

612 

618 

625 

631 

637 

643 

649 

655 

661 

667 

719 

673 

679 

685 

691 

697 

703 

709 

715 

721 

727 

720 

733 

739 

745 

751 

757 

763 

769 

775 

781 

788 

721 

794 

800 

806 

812 

818 

824 

830 

836 

842 

848 

1 

I 

722 

854 

860 

866 

872 

878 

884 

890 

896 

902 

908 

2 

I 

723 

914 

920 

926 

932 

938 

944 

950 

956 

962 

968 

3 

2 

724 

974 

980 

986 

992 

998 

*oo4 

*OIO 

*oi6 

*022 

*028 

4 
5 
6 

2 

3 
4 

725 

86  034 

040 

046 

052 

058 

064 

070 

076 

082 

088 

726 

094 

100 

106 

112 

118 

124 

130 

136 

141 

147 

7 

4 

727 

153 

159 

165 

171 

177 

183 

189 

195 

201 

207 

8 

5 

728 

213 

219 

225 

231 

237 

243 

249 

255 

261 

267 

9 

5 

729 

273 

279 

285 

291 

297 

303 

308 

314 

320 

326 

730 

332 

338 

344 

350 

356 

362 

368 

374 

380 

386 

731 

392 

398 

404 

410 

415 

421 

427 

433 

439 

445 

732 

451 

457 

463 

469 

475 

481 

487 

493 

499 

504 

733 

510 

516 

522 

528 

534 

540 

546 

552 

558 

564 

734 

570 

576 

581 

587 

593 

599 

605 

611 

617 

623 

n 

735 

629 

635 

641 

646 

652 

658 

664 

670 

676 

682 

736 

688 

694 

700 

705 

711 

717 

723 

729 

735 

741 

1 

I 

737 

747 

753 

759 

764 

770 

776 

782 

788 

794 

800 

2 

I 

738 

806 

812 

817 

823 

829 

835 

841 

847 

853 

859 

3 

2 

739 
740 

864 

870 

876 

882 

888 

894 

900 

906 

91T 

917 

4 
5 
6 

2 
3 
3 

923 

929 

935 

941 

947 

953 

958 

964 

970 

976 

741 

982 

988 

994 

999 

*oo5 

*OII 

*oi7 

*023 

*029 

*035 

7 

4 

742 

87  040 

046 

052 

058 

064 

070 

075 

081 

087 

093 

8 
9 

4 

743 

099 

105 

III 

116 

122 

128 

134 

140 

146 

151 

5 

744 

157 

163 

169 

175 

181 

186 

192 

198 

204 

210 

745 

216 

221 

227 

233 

239 

245 

251 

256 

262 

268 

746 

274 

280 

286 

291 

297 

303 

309 

315 

320 

326 

747 

332 

33^ 

344 

349 

355 

361 

367 

373 

379 

384 

748 

390 

396 

402 

408 

413 

419 

425 

431 

437 

442 

749 

448 

454 

460 

466 

471 

477 

483 

489 

495 

500 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

P.P. 

LOGARITHMS   OF   NUMBERS. 


39 


N. 

O 

1 

2 

3 

4 

5 

0 

7 

8 

%} 

P.P. 

750 

87  506 

512 

518 

523 

529 

535 

541 

547 

552 

558 

751 

564 

570 

576 

581 

587 

593 

599 

604 

610 

616 

752 

622 

628 

633 

639 

645 

651 

656 

662 

668 

674 

753 

679 

685 

691 

697 

703 

708 

714 

720 

726 

731 

754 

737 

743 

749 

754 

760 

766 

772 

777 

783 

789 

755 

795 

800 

806 

812 

818 

823 

829 

835 

841 

846 

756 

852 

858 

864 

869 

875 

881 

887 

892 

898 

904 

757 

910 

915 

921 

927 

933 

938 

944 

950 

955 

961 

758 

967 

973 

978 

984 

990 

996 

*OOI 

*oo7 

*oi3 

*oi8 

759 

88  024 

030 

036 

041 

047 

053 

058 

064 

070 

076 

760 

081 

087 

093 

098 

104 

no 

116 

121 

127 

^ZZ 

761 

138 

144 

150 

156 

161 

167 

173 

178 

184 

190 

1 

6 

I 

762 

195 

201 

207 

213 

218 

224 

230 

235 

241 

247 

2 

I 

763 

252 

258 

264 

270 

275 

281 

287 

292 

298 

304 

3 

2 

764 

309 

315 

321 

326 

332 

338 

343 

349 

355 

360 

4 
5 
6 

2 
3 

4 

765 

366 

372 

377 

383 

389 

395 

400 

406 

412 

417 

766 

423 

429 

434 

440 

446 

451 

457 

463 

468 

474 

7 

4 

767 

480 

485 

491 

497 

502 

508 

513 

519 

525 

530 

8 

5 

768 

536 

542 

547 

553 

559 

564 

570 

576 

581 

587 

9 

5  • 

769 

593 

598 

604 

610 

615 

621 

627 

632 

638 

643 

770 

649 

655 

660 

666 

672 

677 

683 

689 

694 

700 

771 

705 

711 

717 

722 

728 

734 

739 

745 

750 

756 

772 

762 

767 

773 

779 

784 

790 

795 

801 

807 

812 

773 

818 

824 

829 

835 

840 

846 

852 

857 

863 

868 

774 

874 

880 

885 

891 

897 

902 

908 

913 

919 

925 

775 

930 

936 

941 

947 

953 

958 

964 

969 

975 

981 

776 

986 

992 

997 

*oo3 

*oo9 

*oi4 

*020 

*025 

*o3i 

*o37 

777 

89  042 

048 

053 

059 

064 

070 

076 

081 

087 

092 

778 

098 

104 

109 

115 

120 

126 

131 

137 

143 

148 

779 

154 

159 

165 

.  170 

176 

182 

187 

193 

198 

204 

780 

209 

215 

221 

226 

232 

237 

243 

248 

254 

260 

781 

265 

271 

276 

282 

287 

293 

298 

304 

310 

315 

1 
2 

6 

782 

321 

326 

332 

337 

343 

348 

354 

360 

365 

371 

783 

376 

382 

387 

393 

398 

404 

409 

415 

421 

426 

3 

2 

784 

432 

437 

443 

448 

454 

459 

465 

470 

476 

481 

4 
5 
6 

3 

7 

785 

487 

492 

498 

504 

509 

515 

520 

526 

531 

537 

786 

542 

548 

553 

559 

564 

570 

575 

581 

586 

592 

7 

4 

787 

597 

603 

609 

614 

620 

625 

631 

636 

642 

647 

8 

4 

788 

653 

658 

664 

669 

675 

680 

686 

691 

697 

702 

9 

5 

789 

708 

713 

719 

724 

730 

735 

741 

746 

752 

757 

790 

763 

768 

774 

779 

785 

790 

796 

801 

807 

812 

791 

818 

823 

829 

834 

840 

845 

851 

856 

862 

867 

792 

873 

878 

883 

889 

894 

900 

905 

911 

916 

922 

793 

927 

933 

938 

944 

949 

955 

960 

966 

971 

977 

794 

982 

988 

993 

998 

*oo4 

*oo9 

*oi5 

*0  20 

*026 

*o3i 

795 

90  037 

042 

048 

053 

059 

064 

069 

075 

080 

086 

796 

091 

097 

102 

108 

113 

119 

124 

129 

135 

140 

797 

146 

151 

157 

162 

168 

173 

179 

184 

189 

195 

798 

200 

206 

211 

217 

222 

227 

^3>3> 

238 

244 

249 

799 

255 

260 

266 

271 

276 

282 

287 

293 

298 

304 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

P.P. 

40 


LOGARITHMS   OF   NUMBERS. 


N. 

O 

1 

«^ 

3 

4 

5 

6 

7 

8 

9 

P.P. 

800 

90  309 

314 

320 

325 

?>?,'' 

i  336 

342 

347 

352 

358 

801 

363 

369 

374 

380 

385 

1  390 

396 

401 

407 

412 

802 

417 

423 

428 

434 

439 

445 

450 

455 

461 

466 

803 

472 

477 

482 

488 

493 

499 

504 

509 

515 

520 

804 

526 

580 

531 

585 

536 
590 

542 
596 

547 
601 

553 

558 

563 

569 

574 

805 

607 

612 

617 

623 

628 

806 

634 

639 

644 

650 

655 

660 

666 

671 

677 

682 

807 

687 

693 

698 

703 

709 

714 

720 

725 

730 

736 

808 

741 

747 

752 

757 

763 

768 

773 

779 

784 

789 

809 

795 

800 

806 

811 

816 

822 

827 

832 

^3^ 

843 

a 

810 

849 

854 

859 

865 

870 

875 

881 

886 

891 

897 

811 

902 

907 

913 

918 

924 

929 

934 

940 

945 

950 

1 

I 

812 

956 

961 

966 

972 

977 

982 

988 

993 

998 

*oo4 

2 

I 

813 

91  009 

014 

020 

025 

030 

036 

041 

046 

052 

057 

3 

2 

814 

062 

068 

073 

078 

084 

089 

094 

100 

105 

no 

4 
5 
6 

2 

3 
4 

815 

116 

121 

126 

132 

137 

142 

148 

153 

158 

164 

816 

169 

174 

180 

185 

190 

196 

201 

206 

212 

217 

7 

4 

817 

222 

228 

^?>2, 

238 

243 

249 

254 

259 

265 

270 

8 
9 

5 

818 

275 

281 

286 

291 

297 

302 

307 

312 

318 

323 

5 

819 

328 

334 

339 

344 

350 

355 

360 

365 

371 

376 

820 

381 

387 

392 

397 

403 

408 

413 

418 

424 

429 

821 

434 

440 

445 

450 

455 

461 

466 

471 

477 

482 

822 

487 

492 

498 

503 

508 

514 

519 

524 

529 

535 

823 

540 

545 

551 

556 

561 

566 

572 

577 

582 

587 

824 

593 

598 

603 

609 

614 

619 

624 

630 

635 

640 

825 

645 

651 

656 

661 

666 

672 

677 

682 

687 

693 

826 

698 

703 

709 

714 

719 

724 

730 

735 

740 

745 

827 

751 

756 

761 

766 

772 

777 

782 

787 

793 

798 

828 

803 

808 

814 

819 

824 

829 

834 

840 

845 

850 

829 

855 

861 

866 

871 

876 

882 

887 

892 

897 

903 

830 

908 

913 

918 

924 

929 

934 

939 

944 

950 

955 

831 

960 

965 

971 

976 

981 

986 

991 

997 

*002 

*oo7 

1 

0 

832 

92  012 

018 

023 

028 

033 

038 

044 

049 

054 

059 

2 

1 

833 

065 

070 

075 

080 

085 

091 

096 

lOI 

106 

III 

3 

2 

834 

117 

122 

127 

^3^ 

137 

143 

148 

153 

158 

163 

4 
5 
6 

2 
3 
3 

835  1 

169 

174 

179 

184 

189 

195 

200 

205 

210 

215 

836 

221 

226 

231 

236 

241 

247 

252 

257 

262 

267 

7 

4 

837 

273 

278 

283 

288 

293 

298 

304 

309 

314 

319 

8 

4 

838  I 

324 

330 

335 

340 

345 

350 

355 

361 

366 

371 

9 

5 

839  1 

376 

381 

387 

392 

397 

402 

407 

412 

418 

423 

840 

428 

433 

438 

443 

449 

454 

459 

464 

469 

474 

841 

480 

485 

490 

495 

500 

505 

511 

516 

521 

526 

842 

531 

536 

542 

547 

552 

557 

562 

567 

572 

578 

843 

583 

588 

593 

598 

603 

609 

614 

619 

624 

629 

844 

845  ! 

634 

639 

645 

650 

655 

660 

665 

670 

675 

681 

686 

691 

696 

701 

706 

711 

716 

722 

727 

732 

846 

737 

742 

747 

752 

758 

763 

768 

773 

778 

783 

847 

788 

793 

799 

804 

809 

814 

819 

824 

829 

834 

848 

840 

845 

850 

855 

860 

865 

870 

875 

881 

886 

849 

891 

896 

901 

906 

911 

916 

921 

927 

932 

937 

N. 

0 

1 

2 

3 

4 

5 

e 

7 

8 

9 

P.P. 

LOGARITHMS   OF   NUMBERS. 


41 


N. 

0 

1 

2 

3 

4 

0 

0 

7 

8 

U 

P. 

P. 

850 

92  942 

947 

952 

957 

962 

967 

973 

978 

983 

988 

851 

993 

998 

*oo3 

*oo8 

*oi3 

*oi8 

*024 

*029 

*034 

*039 

852 

93  044 

049 

054 

059 

064 

069 

075 

080 

085 

090 

853 

095 

100 

105 

no 

115 

120 

125 

131 

136 

141 

854 

146 

151 

156 

161 

166 

171 

176 

181 

186 

192 

6 

I 

855 

197 

202 

207 

212 

217 

222 

227 

232 

237 

242 

856 

247 

252 

258 

263 

268 

273 

278 

283 

288 

293 

1 

857 

298 

303 

308 

^-^^ 

318 

323 

328 

334 

339 

344 

2 

I 

858 

349 

354 

359 

364 

369 

374 

379 

384 

389 

394 

3 

2 

859 

399 

404 

409 

414 

420 

425 

430 

435 

440 

445 

4 
5 
6 

2 
3 
4 

860 

450 

455 

460 

465 

470 

475 

480 

485 

490 

495 

861 

500 

505 

510 

515 

520 

526 

S?,^ 

536 

541 

546 

7 

4 

862 

551 

556 

561 

S66 

571 

576 

581 

S86 

591 

596 

8 
9 

5 

863 

601 

606 

611 

616 

621 

626 

631 

636 

641 

646 

5 

864 

651 

656 

661 

666 

671 

676 

682 

687 

692 

697 

865 

702 

707 

712 

717 

722 

727 

732 

737 

742 

747 

866 

752 

757 

762 

767 

772 

777 

782 

787 

792 

797 

867 

802 

807 

812 

817 

822 

827 

832 

837 

842 

847 

868 

852 

857 

862 

867 

872 

877 

S^2 

887 

892 

897 

869 

902 

907 

912 

917 

922 

927 

932 

937 

942 

947 

K 

870 

952 

957 

962 

967 

972 

977 

982 

987 

992 

997 

871 

94  002 

007 

012 

017 

022 

027 

032 

037 

042 

047 

1 

I 

872 

052 

057 

062 

067 

072 

077 

082 

086 

091 

096 

2 

I 

873 

lOI 

106 

III 

116 

121 

126 

131 

136 

141 

146 

3 

2 

874 

151 

156 

161 

166 

171 

176 

181 

186 

191 

196 

4 
5 
6 

2 
3 

3 

875 

201 

206 

211 

216 

221 

226 

231 

236 

240 

245 

876 

250 

255 

260 

265 

270 

275 

280 

285 

290 

295 

7 

4 

877 

300 

305 

310 

315 

320 

325 

330 

335 

340 

345 

8 

4 

878 

349 

354 

359 

364 

369 

374 

379 

384 

389 

394 

9 

5 

879 

399 

404 

409 

414 

419 

424 

429 

433 

438 

443 

880 

448 

453 

458 

463 

468 

473 

478 

483 

488 

493 

881 

498 

503 

507 

512 

517 

522 

527 

532 

537 

542 

882 

547 

552 

557 

562 

567 

571 

576 

581 

S86 

591 

883 

596 

601 

606 

611 

616 

621 

626 

630 

635 

640 

884 

645 

650 

655 

660 

665 

670 

675 

680 

685 

689 

4 

885 

694 

699 

704 

709 

714 

719 

724 

729 

734 

738 

886 

743 

748 

753 

758 

763 

768 

773 

778 

783 

787 

1 

0 

887 

792 

797 

802 

807 

812 

817 

822 

827 

832 

836 

2 

I 

888 

841 

846 

851 

856 

861 

866 

871 

876 

880 

885 

3 

I 

889 

890 

«95 

900 

905 

910 

915 

919 

924 

929 

934 

4 
5 
6 

2 
2 
2 

890 

939 

944 

949 

954 

959 

963 

968 

973 

978 

983 

891 

988 

993 

998 

*002 

*oo7 

*OI2 

*oi7 

*022 

*027 

^032 

7 

3 

892 

95  036 

041 

046 

051 

056 

061 

066 

071 

075 

080 

8 
9 

3 

893 

085 

090 

095 

100 

105 

109 

114 

119 

124 

129 

4 

894 

134 

139 

143 

148 

153 

158 

163 

168 

173 

177 

895 

182 

187 

192 

197 

202 

207 

211 

216 

221 

226 

896 

231 

236 

240 

245 

250 

255 

260 

265 

270 

274 

897 

279 

284 

289 

294 

299 

3OZ 

308 

3^Z 

318 

S^3 

898 

328 

3Z^ 

Z?>1 

342 

347 

352 

357 

361 

366 

371 

899 

376 

3«i 

386 

390 

395 

400 

405 

410 

415 

419 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

1> 

P 

P. 

42 


LOGARITHMS  OF    NUMBERS. 


N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

P.P. 

900 

95  424 

429 

434 

439 

444 

448 

453 

458 

463 

468 

901 

472 

477 

482 

487 

492 

497 

501 

506 

511 

516 

902 

521 

525 

530 

535 

540 

545 

550 

554 

559 

564 

903 

569 

574 

578 

583 

588 

593 

598 

602 

607 

612 

904 

617 

622 

626 

631 

636 

641 

646 

650 

655 

660 

905 

665 

670 

674 

679 

684 

689 

694 

698 

703 

708 

906 

713 

718 

722 

727 

732 

737 

742 

746 

751 

756 

907 

761 

766 

770 

775 

780 

785 

789 

794 

799 

804 

908 

809 

813 

818 

823 

828 

832 

837 

842 

847 

852 

909 

856 

861 

866 

871 

875 

880 
928 

885 
933 

890 
938 

895 
942 

899 
947 

K 

910 

904 

909 

914 

918 

923 

911 

952 

957 

961 

966 

971 

976 

980 

985 

990 

995 

1 

I 

912 

999 

*oo4 

*oo9 

*oi4 

*oi9 

*023 

*028 

*033 

*038 

*042 

2 

I 

913 

96  047 

052 

057 

061 

066 

071 

076 

080 

085 

090 

3 

2 

914 

095 

099 

104 

109 

114 

118 

123 

128 

^ZZ 

^Z1 

4 
5 
6 

2 

3  ■ 
3 

■915 

142 

147*152 

156 

161 

166 

171 

175 

180 

185 

916 

190 

194 

199 

204 

209 

213 

218 

223 

227 

232 

7 

4 

:917 

^Zl 

242 

246 

^S"^ 

256 

261 

265 

270 

275 

280 

8 
9 

4 

918 

284 

289 

294 

298 

303 

308 

Z^Z 

317 

322 

327 

5 

919 

332 

?>Z^ 

341 

346 

350 

355 

360 

365 

369 

374 

920 

379 

384 

388 

393 

398 

402 

407 

412 

417 

421 

921 

426 

431 

43  5 

440 

445 

450 

454 

459 

464 

468 

922 

473 

478 

483 

487 

492 

497 

501 

506 

511 

515 

923 

520 

525 

530 

534 

539 

544 

548 

553 

558 

562 

924 

567 

572 

577 

581 

586 

591 

595 

600 

605 

609 

925 

614 

619 

624 

628 

633 

638 

642 

647 

652 

656 

926 

661 

666 

670 

675 

680 

685 

689 

694 

699 

703 

927 

708 

713 

717 

722 

727 

731 

736 

741 

745 

750 

928 

755 

759 

764 

769 

774 

778 

783 

788 

792 

797 

929 

802 

806 

811 

816 

820 

825 

830 

834 

839 

844 

930 

848 

853 

858 

862 

867 

872 

876 

881 

886 

890 

931 

895 

900 

904 

909 

914 

918 

923 

928 

932 

937 

1 

0 

932 

942 

946 

951 

956 

960 

965 

970 

974 

979 

984 

2 

933 

988 

993 

997 

*002 

*oo7 

*OII 

*oi6 

*02I 

"^025 

*030 

3 

I 

934 

97  035 

039 

044 

049 

053 

058 

063 

067 

072 

077 

4 
5 
6 

2 
2 
2 

935 

081 

086 

090 

095 

100 

104 

109 

114 

118 

123 

936 

128 

132 

137 

142 

146 

151 

155 

160 

165 

169 

7 

3 

937 

174 

179 

183 

188 

192 

197 

202 

206 

211 

216 

8 

3 

938 

220 

225 

230 

234 

239 

243 

248 

253 

257 

262 

9 

4 

939 

267 

271 

276 

280 

285 

290 

294_ 

299 

304 

308 

940 

z-'z 

317 

322 

327 

ZZ"^ 

ZZ^ 

340 

345 

350 

354 

941 

359 

364 

368 

373 

377 

382 

387 

391 

39<^ 

400 

942 

405 

410 

414 

419 

424 

428 

433 

437 

442 

447 

943 

451 

456 

460 

465 

470 

474 

479 

483 

488 

493 

944 

497 

502 

506 

511 

516 

520 

525 

529 

534 

539 

945 

543 

548 

552 

557 

562 

566 

571 

575 

580 

585 

946 

589 

594 

598 

603 

607 

612 

617 

621 

626 

630 

947 

635 

640 

644 

649 

653 

658 

663 

667 

672 

676 

948 

681 

685 

690 

695 

699 

704 

708 

713 

717 

722 

949 

727 

731 

736 

740 

745 

749 

754 

759 

763 

768 

N. 

0 

1 

2 

3 

4 

5 

G 

7 

8 

9 

P.P. 

LOGARITHMS  OF  NUMBRBB. 


43 


N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

P.P. 

950 

97  772 

777 

782 

786 

791 

795 

800 

804 

809 

813 

951 

8i8 

823 

827 

832 

836 

841 

845 

850 

855 

859 

952 

864 

868 

873 

877 

882 

886 

891 

896 

900 

905 

953 

909 

914 

918 

923 

928 

932 

937 

941 

946 

950 

954 

955 

959 

964 

968 

973 

978 
023 

982 

028 

987 
032 

991 
037 

996 

041 

955 

98  000 

005 

009 

014 

019 

956 

046 

050 

055 

059 

064 

068 

073 

078 

082 

087 

957 

091 

096 

100 

105 

109 

114 

118 

^23 

127 

132 

958 

137 

141 

146 

150 

155 

159 

164 

168 

^73 

177 

959 

182 

186 

191 

195 

200 

204 

209 

214 

218 

223 

R 

960 

227 

232 

236 

241 

245 

250 

254 

259 

263 

268 

961 

272 

277 

281 

286 

290 

295 

299 

304 

308 

3^3 

1 

I 

962 

318 

322 

327 

^S^ 

336 

340 

345 

349 

354 

358 

2 

I 

963 

363 

367 

372 

376 

3«i 

3^S 

390 

394 

399 

403 

3 

2 

964 

408 

412 

417 

421 

426 

430 

435 

439 

444 

448 

4 
5 
6 

2 
3 
3 

965 

453 

457 

462 

466 

471 

475 

480 

484 

489 

493 

966 

498 

502 

507 

511 

516 

520 

525 

529 

534 

538 

7 

4 

967 

543 

547 

552 

556 

561 

565 

570 

574 

579 

583 

8 
9 

4 

968 

588 

592 

597 

601 

605 

610 

614 

619 

623 

628 

5 

969 

632 

637 

641 

646 

650 

65s 

659 

664 

668 

673 

970 

677 

682 

686 

691 

695 

700 

704 

709 

713 

717 

971 

722 

726 

731 

735 

740 

744 

749 

753 

758 

762 

972 

767 

771 

776 

780 

784 

789 

793 

798 

802 

807 

973 

811 

816 

820 

825 

829 

834 

838 

843 

847 

851 

974 

856 

860 

865 

869 

874 

878 

883 

887 

892 

896 

975 

900 

905 

909 

914 

918 

923 

927 

932 

936 

941 

976 

945 

949 

954 

958 

963 

967 

972 

976 

981 

985 

977 

989 

994 

998 

*oo3 

*oo7 

*OI2 

*oi6 

*02I 

*025 

*029 

978 

99  034 

038 

043 

047 

052 

056 

061 

065 

069 

074 

979 

078 

083 

087 

092 

096 

100 

105 

109 

114 

118 

980 

123 

127 

131 

136 

140 

145 

149 

154 

158 

162 

981 

167 

171 

176 

180 

185 

189 

193 

198 

202 

207 

1 

4 

982 

211 

216 

220 

224 

229 

233 

238 

242 

247 

251 

2 

983 

255 

260 

264 

269 

273 

277 

282 

286 

291 

295 

3 

1 

984 

300 

304 

308 

3^3 

317 

322 

326 

330 

335 

339 

4 
5 
6 

2 
2 
2 

985 

344 

348 

352 

357 

361 

366 

370 

374 

379 

383 

986 

388 

392 

396 

401 

405 

410 

414 

419 

423 

427 

7 

3 

987 

432 

436 

441 

445 

449 

454 

458 

463 

467 

471 

8 

3 

988 

476 

480 

484 

489 

493 

498 

502 

506 

511 

515 

9 

4 

989 

520 

524 

528 

533 

537 

542 

546 

550 

555 

559 

990 

564 

S68 

572 

577 

581 

585 

590 

594 

599 

603 

991 

607 

612 

616 

621 

625 

629 

634 

638 

642 

647 

992 

651 

656 

660 

664 

669 

673 

677 

682 

686 

691 

993 

695 

699 

704 

708 

712 

717 

721 

726 

730 

734 

994 

739 

743 

747 

752 

756 

760 

765 

769 

774 

778 

995 

782 

787 

791 

795 

800 

804 

808 

813 

817 

822 

996 

826 

830 

835 

839 

843 

848 

852 

856 

861 

865 

997 

870 

874 

878 

88s 

887 

891 

896 

900 

904 

909 

998 

913 

917 

922 

926 

930 

935 

939 

944 

948 

952 

999 

957 

961 

965 

970 

974 

978 

983 

987 

991 

996 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

P.P. 

1 

.OOf 


3 
4 

.60 


5 

.6' 

6 

.7' 


.8 

8 

90J 


UNIVERSITY    OF    CALIFORNIA 
BRANCH    OF    THE    COLLEGE    OF    AGRICULTURE 


THIS  BOOK  IS  DUE  ON  THE  LAST  DATE 
STAMPED  BELOW 


JH.- 


'M^ 


:;V»w5. 


«     e  T  Nvr* 

■*i 

DEC  1  6  1966 

NOV  2     W. 
J^N  2  4  f9^ 

IWH  2  41964 
FEB  1  0  1964 

7  mj 

JUN  3  3  m'T 

5EP18'.SB7 
DEC  1  6  1957 

DEC  1 3  RS?B- 

m,  3  0 I9S9 

^m  30  REC'fl 

SEP*-''* 

JAN  2  2196* 

\ 

RF?."^'fMM1-;SS^ggf 

>                                     6iii-8,'26 

/ 


I 


UNIVERSITY  OF  CAUFORNIA  UBRARY 


